text: 19 Kraus S, Monnier JD, Ireland MJ, Duchˆene G, Espaillat C, H¨onig S, Juhasz A, Mordasini C, Olofsson J, Paladini C, Stassun K, Turner N, Vasisht G, Harries TJ, Bate MR, Gonzalez JF, Matter A, Zhu Z, Panic O, Regaly Z, Morbidelli A, Meru F, Wolf S, Ilee J, Berger JP, Zhao M, Kral Q, Morlok A, Bonsor A, Ciardi D, Kane SR, Kratter K, Laughlin G, Pepper J, Raymond S, Labadie L, Nelson RP, Weigelt G, ten Brummelaar T, Pierens A, Oudmaijer R, Kley W, Pope B, Jensen ELN, Bayo A, Smith M, Boyajian T, Quiroga- Nun˜ez LH, Millan-Gabet R, Chiavassa A, Gallenne A, Reynolds M, de Wit WJ, Wittkowski M, Millour F, Gandhi P, Ramos Almeida C, Alonso Herrero A, Packham C, Kishimoto M, Tristram KRW, Pott JU, Surdej J, Buscher D, Haniﬀ C, Lacour S, Petrov R, Ridgway S, Tuthill P, van Belle G, Armitage P, Baruteau C, Benisty M, Bitsch B, Paardekooper SJ, Pinte C, Masset F, Rosotti G (2016) Planet Formation Imager (PFI): science vision and key requirements. In: Optical and Infrared Interferometry and Imaging V, Proc SPIE, vol 9907, p 99071K, DOI 10.1117/12.2231067, 1608.00578 Lacour S, Tuthill P, Monnier JD, Kotani T, Gauchet L, Labeye P (2014) A new interferometer architecture combining nulling with phase closure mea- surements. MNRAS439:4018–4029, DOI 10.1093/mnras/stu258, 1306.5184 Le Bouquin JB, Absil O (2012) On the sensitivity of closure phases to faint companions in optical long baseline interferometry. A&A541:A89, DOI 10. 1051/0004-6361/201117891, 1204.3721 Le Bouquin JB, Rousselet-Perraut K, Berger JP, Herwats E, Benisty M, Ab- sil O, Defrere D, Monnier J, Traub W (2008) Polar-interferometry: what can be learnt from the IOTA/IONIC experiment. In: Optical and Infrared Interferometry, Proc SPIE, vol 7013, p 70130F, DOI 10.1117/12.786377 Le Bouquin JB, Berger JP, Lazareﬀ B, Zins G, Haguenauer P, Jocou L, Kern P, Millan-Gabet R, Traub W, Absil O, Augereau JC, Benisty M, Blind N, Bonﬁls X, Bourget P, Delboulbe A, Feautrier P, Germain M, Gitton P, Gillier D, Kiekebusch M, Kluska J, Knudstrup J, Labeye P, Lizon JL, Monin JL, Magnard Y, Malbet F, Maurel D, M´enard F, Micallef M, Michaud L, Montagnier G, Morel S, Moulin T, Perraut K, Popovic D, Rabou P, Rochat S, Rojas C, Roussel F, Roux A, Stadler E, Steﬂ S, Tatulli E, Ventura N (2011) PIONIER: a 4-telescope visitor instrument at VLTI. A&A535:A67, DOI 10.1051/0004-6361/201117586, 1109.1918 L´ena P, Absil O, Borkowski V, Herwats E, D M, S Q, P R (2005) 37th Li Inter- national Astrophysics Colloquium : conclusions and perspectives. Bulletin de la Soci Royale des Sciences de Li`ege 74:203–229 Lopez B, Lagarde S, Jaﬀe W, Petrov R, Sch¨oller M, Antonelli P, Beckmann U, Berio P, Bettonvil F, Glindemann A, Gonzalez JC, Graser U, Hofmann KH, Millour F, Robbe-Dubois S, Venema L, Wolf S, Henning T, Lanz T, Weigelt G, Agocs T, Bailet C, Bresson Y, Bristow P, Dugu´e M, Heininger M, Kroes G, Laun W, Lehmitz M, Neumann U, Augereau JC, Avila G, Behrend J, van Belle G, Berger JP, van Boekel R, Bonhomme S, Bourget P, Brast R, Clausse JM, Connot C, Conzelmann R, Cruzal`ebes P, Csepany G, Danchi W, Delbo M, Delplancke F, Dominik C, van Duin A, Elswijk E, Fantei Y, Finger G, Gabasch A, Gay J, Girard P, Girault V, Gitton P, ##LTLine##

text: Kraus

label: reference

text: S,

text: Monnier

text: JD,

text: Ireland

text: MJ,

text: Duchˆene

text: G,

text: Espaillat

text: C,

text: H¨onig

text: S,

text: Juhasz

text: A,

text: Mordasini

text: C,

text: Olofsson

text: J,

text: Paladini

text: C,

text: Stassun

text: K,

text: Turner

text: N,

text: Vasisht

text: G,

text: Harries

text: TJ,

text: Bate

text: MR,

text: Gonzalez

text: JF,

text: Matter

text: A,

text: Zhu

text: Z,

text: Panic

text: O,

text: Regaly

text: Z,

text: Morbidelli

text: A,

text: Meru

text: F,

text: Wolf

text: S,

text: Ilee

text: J,

text: Berger

text: JP,

text: Zhao

text: M,

text: Kral

text: Q,

text: Morlok

text: A,

text: Bonsor

text: A,

text: Ciardi

text: D,

text: Kane

text: SR,

text: Kratter

text: K,

text: Laughlin

text: G,

text: Pepper

text: J,

text: Raymond

text: S,

text: Labadie

text: L,

text: Nelson

text: RP,

text: Weigelt

text: G,

text: ten

text: Brummelaar

text: T,

text: Pierens

text: A,

text: Oudmaijer

text: R,

text: Kley

text: W,

text: Pope

text: B,

text: Jensen

text: ELN,

text: Bayo

text: A,

text: Smith

text: M,

text: Boyajian

text: T,

text: Quiroga-

text: Nun˜ez

text: LH,

text: Millan-Gabet

text: R,

text: Chiavassa

text: A,

text: Gallenne

text: A,

text: Reynolds

text: M,

text: de

text: Wit

text: WJ,

text: Wittkowski

text: M,

text: Millour

text: F,

text: Gandhi

text: P,

text: Ramos

text: Almeida

text: C,

text: Alonso

text: Herrero

text: A,

text: Packham

text: C,

text: Kishimoto

text: M,

text: Tristram

text: KRW,

text: Pott

text: JU,

text: Surdej

text: J,

text: Buscher

text: D,

text: Haniﬀ

text: C,

text: Lacour

text: S,

text: Petrov

text: R,

text: Ridgway

text: S,

text: Tuthill

text: P,

text: van

text: Belle

text: G,

text: Armitage

text: P,

text: Baruteau

text: C,

text: Benisty

text: M,

text: Bitsch

text: B,

text: Paardekooper

text: SJ,

text: Pinte

text: C,

text: Masset

text: F,

text: Rosotti

text: G

text: Planet

text: Formation

text: Imager

text: (PFI):

text: science

text: vision

text: and

text: key

text: requirements.

text: In:

text: Optical

text: and

text: Infrared

text: Interferometry

text: and

text: Imaging

text: V,

text: Proc

text: SPIE,

text: vol

text: p

text: DOI

text: Lacour

text: S,

text: Tuthill

text: P,

text: Monnier

text: JD,

text: Kotani

text: T,

text: Gauchet

text: L,

text: Labeye

text: P

text: A

text: new

text: interferometer

text: architecture

text: combining

text: nulling

text: with

text: phase

text: closure

text: mea-

text: surements.

text: DOI

text: Le

text: Bouquin

text: JB,

text: Absil

text: O

text: On

text: the

text: sensitivity

text: of

text: closure

text: phases

text: to

text: faint

text: companions

text: in

text: optical

text: long

text: baseline

text: interferometry.

text: DOI

text: Le

text: Bouquin

text: JB,

text: Rousselet-Perraut

text: K,

text: Berger

text: JP,

text: Herwats

text: E,

text: Benisty

text: M,

text: Ab-

text: sil

text: O,

text: Defrere

text: D,

text: Monnier

text: J,

text: Traub

text: W

text: Polar-interferometry:

text: what

text: can

text: be

text: learnt

text: from

text: the

text: IOTA/IONIC

text: experiment.

text: In:

text: Optical

text: and

text: Infrared

text: Interferometry,

text: Proc

text: SPIE,

text: vol

text: p

text: DOI

text: Le

text: Bouquin

text: JB,

text: Berger

text: JP,

text: Lazareﬀ

text: B,

text: Zins

text: G,

text: Haguenauer

text: P,

text: Jocou

text: L,

text: Kern

text: P,

text: Millan-Gabet

text: R,

text: Traub

text: W,

text: Absil

text: O,

text: Augereau

text: JC,

text: Benisty

text: M,

text: Blind

text: N,

text: Bonﬁls

text: X,

text: Bourget

text: P,

text: Delboulbe

text: A,

text: Feautrier

text: P,

text: Germain

text: M,

text: Gitton

text: P,

text: Gillier

text: D,

text: Kiekebusch

text: M,

text: Kluska

text: J,

text: Knudstrup

text: J,

text: Labeye

text: P,

text: Lizon

text: JL,

text: Monin

text: JL,

text: Magnard

text: Y,

text: Malbet

text: F,

text: Maurel

text: D,

text: M´enard

text: F,

text: Micallef

text: M,

text: Michaud

text: L,

text: Montagnier

text: G,

text: Morel

text: S,

text: Moulin

text: T,

text: Perraut

text: K,

text: Popovic

text: D,

text: Rabou

text: P,

text: Rochat

text: S,

text: Rojas

text: C,

text: Roussel

text: F,

text: Roux

text: A,

text: Stadler

text: E,

text: Steﬂ

text: S,

text: Tatulli

text: E,

text: Ventura

text: N

text: PIONIER:

text: a

text: visitor

text: instrument

text: at

text: VLTI.

text: DOI

text: L´ena

text: P,

text: Absil

text: O,

text: Borkowski

text: V,

text: Herwats

text: E,

text: D

text: M,

text: S

text: Q,

text: P

text: R

text: Li

text: Inter-

text: national

text: Astrophysics

text: Colloquium

text: conclusions

text: and

text: perspectives.

text: Bulletin

text: de

text: la

text: Soci

text: Royale

text: des

text: Sciences

text: de

text: Li`ege

text: Lopez

text: B,

text: Lagarde

text: S,

text: Jaﬀe

text: W,

text: Petrov

text: R,

text: Sch¨oller

text: M,

text: Antonelli

text: P,

text: Beckmann

text: U,

text: Berio

text: P,

text: Bettonvil

text: F,

text: Glindemann

text: A,

text: Gonzalez

text: JC,

text: Graser

text: U,

text: Hofmann

text: KH,

text: Millour

text: F,

text: Robbe-Dubois

text: S,

text: Venema

text: L,

text: Wolf

text: S,

text: Henning

text: T,

text: Lanz

text: T,

text: Weigelt

text: G,

text: Agocs

text: T,

text: Bailet

text: C,

text: Bresson

text: Y,

text: Bristow

text: P,

text: Dugu´e

text: M,

text: Heininger

text: M,

text: Kroes

text: G,

text: Laun

text: W,

text: Lehmitz

text: M,

text: Neumann

text: U,

text: Augereau

text: JC,

text: Avila

text: G,

text: Behrend

text: J,

text: van

text: Belle

text: G,

text: Berger

text: JP,

text: van

text: Boekel

text: R,

text: Bonhomme

text: S,

text: Bourget

text: P,

text: Brast

text: R,

text: Clausse

text: JM,

text: Connot

text: C,

text: Conzelmann

text: R,

text: Cruzal`ebes

text: P,

text: Csepany

text: G,

text: Danchi

text: W,

text: Delbo

text: M,

text: Delplancke

text: F,

text: Dominik

text: C,

text: van

text: Duin

text: A,

text: Elswijk

text: E,

text: Fantei

text: Y,

text: Finger

text: G,

text: Gabasch

text: A,

text: Gay

text: J,

text: Girard

text: P,

text: Girault

text: V,

text: Gitton

text: P,

text: ##LTLine##


text: 5 photometric precision make these measurements very diﬃcult. With directly imaged planets, this method will show its full potential. Although the inner working angle will limit the access to the smallest phase angles, only a mod- ∼ erate photometric precision of 10% will be required to achieve a crucial di- agnostic and ﬁrst classiﬁcation of the targets. Another strength of the method comes from the favorable distribution of orbit orientations. For the median ◦ inclination of randomly oriented orbits (60 ), the amplitude of phase curves is only decreased by 10 % compared with the maximum variations, reached for ◦ a 90 inclination (Maurin et al 2012). Only a few observations covering one orbit would be suﬃcient to start deriving constraints on the climate and sev- eral low-resolution or broadband observations would be necessary to resolve the orbital motion of the planet. 2.2 Surface conditions, atmospheric composition, and habitability In addition to the constraints from orbital broadband photometry, mid- infrared spectroscopy is fundamental to investigate the nature of planetary atmospheres by detecting spectral features, constraining the temperature and pressure structure, the cloudiness and by determining whether the planet could potentially be habitable. Indeed, several important species have mid-infrared spectral signatures that can be detected at low to medium spectral resolving power (e.g., H O, CO , O , CH , N O, see Table 1). In addition, surface con- 2 2 3 4 2 ditions (temperature and pressure) can be characterised relatively well from ∼ mid-infrared observations (to within 10 K at 3-σ) with S/Ns between 10 and 30, depending on spectral resolution (von Paris et al 2013). By observing a large number of Habitable Zone (HZ) rocky exoplanets, it will be possible to correlate the concept of habitability with key parameters and processes like spectral type of the parent star, degree of stellar activity, the temper- ature/pressure structure of the atmosphere, gaseous composition, the circu- lation and heat transfer of the atmosphere, the atmospheric chemistry and photochemistry, and the outgassing of atmospheric species. 2.3 Understanding the concept of biosignature A biosignature can be deﬁned as “an observable feature of a planet, such as its atmospheric composition, that our present models cannot reproduce when including the abiotic physical and chemical processes we know about” (L´eger et al 2011). The problem with this deﬁnition is that our theoretical models are limited and the emerging vision in the community is that studying the atmo- spheres of a large number of exoplanets will be required to provide an essen- tial context for interpreting possible detections of bio-signatures. A good ap- proach could be to create the planetary equivalent of a “Hertzsprung-Russel“ including species suggestive of life. Only after observing a suﬃcient number of planetary atmospheres, will it be possible to identify possible anomalies in this diagram that cannot be explained without the presence of life. To make progress in that direction, the mid-infrared regime has a key role to play since it contains spectral signatures of important molecules such as H O, CO , O , 2 2 3 N O, and CH . In fact, the triple signature (i.e., O , CO , H O) was con- 2 4 3 2 2 ##LTLine##

label: paragraph

text: photometric

text: precision

text: make

text: these

text: measurements

text: very

text: diﬃcult.

text: With

text: directly

text: imaged

text: planets,

text: this

text: method

text: will

text: show

text: its

text: full

text: potential.

text: Although

text: the

text: inner

text: working

text: angle

text: will

text: limit

text: the

text: access

text: to

text: the

text: smallest

text: phase

text: angles,

text: only

text: a

text: mod-

text: erate

text: photometric

text: precision

text: of

text: will

text: be

text: required

text: to

text: achieve

text: a

text: crucial

text: di-

text: agnostic

text: and

text: ﬁrst

text: classiﬁcation

text: of

text: the

text: targets.

text: Another

text: strength

text: of

text: the

text: method

text: comes

text: from

text: the

text: favorable

text: distribution

text: of

text: orbit

text: orientations.

text: For

text: the

text: median

text: inclination

text: of

text: randomly

text: oriented

text: orbits

text: the

text: amplitude

text: of

text: phase

text: curves

text: is

text: only

text: decreased

text: by

text: compared

text: with

text: the

text: maximum

text: variations,

text: reached

text: for

text: a

text: inclination

text: (Maurin

text: et

text: al

text: Only

text: a

text: few

text: observations

text: covering

text: one

text: orbit

text: would

text: be

text: suﬃcient

text: to

text: start

text: deriving

text: constraints

text: on

text: the

text: climate

text: and

text: sev-

text: eral

text: low-resolution

text: or

text: broadband

text: observations

text: would

text: be

text: necessary

text: to

text: resolve

text: the

text: orbital

text: motion

text: of

text: the

text: planet.

text: Surface

text: conditions,

text: atmospheric

text: composition,

text: and

text: habitability

text: In

text: addition

text: to

text: the

text: constraints

text: from

text: orbital

text: broadband

text: photometry,

text: mid-

text: infrared

text: spectroscopy

text: is

text: fundamental

text: to

text: investigate

text: the

text: nature

text: of

text: planetary

text: atmospheres

text: by

text: detecting

text: spectral

text: features,

text: constraining

text: the

text: temperature

text: and

text: pressure

text: structure,

text: the

text: cloudiness

text: and

text: by

text: determining

text: whether

text: the

text: planet

text: could

text: potentially

text: be

text: habitable.

text: Indeed,

text: several

text: important

text: species

text: have

text: mid-infrared

text: spectral

text: signatures

text: that

text: can

text: be

text: detected

text: at

text: low

text: to

text: medium

text: spectral

text: resolving

text: power

text: (e.g.,

text: H

text: O,

text: CO

text: O

text: CH

text: N

text: O,

text: see

text: Table

text: In

text: addition,

text: surface

text: con-

text: ditions

text: (temperature

text: and

text: pressure)

text: can

text: be

text: characterised

text: relatively

text: well

text: from

text: mid-infrared

text: observations

text: (to

text: within

text: K

text: at

text: with

text: S/Ns

text: between

text: and

text: depending

text: on

text: spectral

text: resolution

text: (von

text: Paris

text: et

text: al

text: By

text: observing

text: a

text: large

text: number

text: of

text: Habitable

text: Zone

text: (HZ)

text: rocky

text: exoplanets,

text: it

text: will

text: be

text: possible

text: to

text: correlate

text: the

text: concept

text: of

text: habitability

text: with

text: key

text: parameters

text: and

text: processes

text: like

text: spectral

text: type

text: of

text: the

text: parent

text: star,

text: degree

text: of

text: stellar

text: activity,

text: the

text: temper-

text: ature/pressure

text: structure

text: of

text: the

text: atmosphere,

text: gaseous

text: composition,

text: the

text: circu-

text: lation

text: and

text: heat

text: transfer

text: of

text: the

text: atmosphere,

text: the

text: atmospheric

text: chemistry

text: and

text: photochemistry,

text: and

text: the

text: outgassing

text: of

text: atmospheric

text: species.

text: Understanding

text: the

text: concept

text: of

text: biosignature

text: A

text: biosignature

text: can

text: be

text: deﬁned

text: as

text: “an

text: observable

text: feature

text: of

text: a

text: planet,

text: such

text: as

text: its

text: atmospheric

text: composition,

text: that

text: our

text: present

text: models

text: cannot

text: reproduce

text: when

text: including

text: the

text: abiotic

text: physical

text: and

text: chemical

text: processes

text: we

text: know

text: about”

text: (L´eger

text: et

text: al

text: The

text: problem

text: with

text: this

text: deﬁnition

text: is

text: that

text: our

text: theoretical

text: models

text: are

text: limited

text: and

text: the

text: emerging

text: vision

text: in

text: the

text: community

text: is

text: that

text: studying

text: the

text: atmo-

text: spheres

text: of

text: a

text: large

text: number

text: of

text: exoplanets

text: will

text: be

text: required

text: to

text: provide

text: an

text: essen-

text: tial

text: context

text: for

text: interpreting

text: possible

text: detections

text: of

text: bio-signatures.

text: A

text: good

text: ap-

text: proach

text: could

text: be

text: to

text: create

text: the

text: planetary

text: equivalent

text: of

text: a

text: “Hertzsprung-Russel“

text: including

text: species

text: suggestive

text: of

text: life.

text: Only

text: after

text: observing

text: a

text: suﬃcient

text: number

text: of

text: planetary

text: atmospheres,

text: will

text: it

text: be

text: possible

text: to

text: identify

text: possible

text: anomalies

text: in

text: this

text: diagram

text: that

text: cannot

text: be

text: explained

text: without

text: the

text: presence

text: of

text: life.

text: To

text: make

text: progress

text: in

text: that

text: direction,

text: the

text: mid-infrared

text: regime

text: has

text: a

text: key

text: role

text: to

text: play

text: since

text: it

text: contains

text: spectral

text: signatures

text: of

text: important

text: molecules

text: such

text: as

text: H

text: O,

text: CO

text: O

text: N

text: O,

text: and

text: CH

text: In

text: fact,

text: the

text: triple

text: signature

text: (i.e.,

text: O

text: CO

text: H

text: O)

text: was

text: con-

text: ##LTLine##


text: 7 Three-Telescope Nuller (TTN, Karlsson et al 2004). These studies included the launch requirements, payload spacecraft, and the ground segment during which the actual mission science would be executed. Almost simultaneously, NASA/JPL initiated a similar study for TPF-I and focused in particular on the Dual-chopped Bracewell (Lay 2004) and X-array conﬁgurations. These eﬀorts on both sides of the Atlantic have ﬁnally resulted in a convergence and con- sensus on mission architecture, the so-called non-coplanar (aka Emma-type) X-array. The baseline design consisted in four collector spacecrafts, ﬂying in rectangular formation and feeding light to the beam combiner spacecraft lo- cated approximately 1200 m from the array. This arrangement makes available baselines up to 170 m for nulling measurements and up to 500 m for the gen- eral astrophysics program (constructive imaging with an angular resolution of 4 mas at 10 µm). 3.3 Exoplanet yield The number of planetary atmospheres that can be studied with such an instrument is a critical metric to estimate the science return of the mission. During the Darwin/TPF studies, the exoplanet yield has been extensively studied and cross-validated between ESA and NASA using various assump- tions on the existing exoplanet population and prevalence of exozodiacal dust, which were both unknown at that time (e.g., Defr`ere et al 2010). Today, the occurrence of HZ rocky exoplanets has been measured by Kepler (e.g., Winn and Fabrycky 2015) and the prevalence of exozodiacal dust has been very well constrained by ground-based nulling interferometers (Mennesson et al 2014, Ertel et al. in prep). It is therefore possible to predict the exoplanet yield of direct imaging instruments more precisely. Recently, based on planet occur- rence statistics from Kepler and using Monte-Carlo simulations, Kammerer ∼ and Quanz (2017) estimated that 315+−113 exoplanets with radii between ∼77 0.5 R⊕ and 6 R⊕ can be detected during 0.52 years of mission time assuming four 2-m apertures and throughputs 3.5 times worse than those for the JWST and 40% overheads. Approximately 85 planets could be habitable (radii be- tween 0.5 R⊕ and 1.75 R⊕ and equilibrium temperatures between 200 K and 450 K) and would be prime targets for follow-up spectroscopic observations. While this exoplanet yield can be interpreted as an upper limit because no ex- ozodiacal dust was assumed in this study, new results from the NASA’s Hunt for Observable Signatures of Terrestrial planetary Systems (HOSTS, Danchi et al 2014) survey on the Large Binocular Telescope Interferometer (HOSTS, Hinz et al 2016) suggest that exozodiacal dust disks would not be a major ##LTLine##

text: Three-Telescope

text: Nuller

text: (TTN,

text: Karlsson

text: et

text: al

text: These

text: studies

text: included

text: the

text: launch

text: requirements,

text: payload

text: spacecraft,

text: and

text: the

text: ground

text: segment

text: during

text: which

text: the

text: actual

text: mission

text: science

text: would

text: be

text: executed.

text: Almost

text: simultaneously,

text: NASA/JPL

text: initiated

text: a

text: similar

text: study

text: for

text: TPF-I

text: and

text: focused

text: in

text: particular

text: on

text: the

text: Dual-chopped

text: Bracewell

text: (Lay

text: and

text: X-array

text: conﬁgurations.

text: These

text: eﬀorts

text: on

text: both

text: sides

text: of

text: the

text: Atlantic

text: have

text: ﬁnally

text: resulted

text: in

text: a

text: convergence

text: and

text: con-

text: sensus

text: on

text: mission

text: architecture,

text: the

text: so-called

text: non-coplanar

text: (aka

text: Emma-type)

text: X-array.

text: The

text: baseline

text: design

text: consisted

text: in

text: four

text: collector

text: spacecrafts,

text: ﬂying

text: in

text: rectangular

text: formation

text: and

text: feeding

text: light

text: to

text: the

text: beam

text: combiner

text: spacecraft

text: lo-

text: cated

text: approximately

text: m

text: from

text: the

text: array.

text: This

text: arrangement

text: makes

text: available

text: baselines

text: up

text: to

text: m

text: for

text: nulling

text: measurements

text: and

text: up

text: to

text: m

text: for

text: the

text: gen-

text: eral

text: astrophysics

text: program

text: (constructive

text: imaging

text: with

text: an

text: angular

text: resolution

text: of

text: mas

text: at

text: µm).

text: Exoplanet

text: yield

text: The

text: number

text: of

text: planetary

text: atmospheres

text: that

text: can

text: be

text: studied

text: with

text: such

text: an

text: instrument

text: is

text: a

text: critical

text: metric

text: to

text: estimate

text: the

text: science

text: return

text: of

text: the

text: mission.

text: During

text: the

text: Darwin/TPF

text: studies,

text: the

text: exoplanet

text: yield

text: has

text: been

text: extensively

text: studied

text: and

text: cross-validated

text: between

text: ESA

text: and

text: NASA

text: using

text: various

text: assump-

text: tions

text: on

text: the

text: existing

text: exoplanet

text: population

text: and

text: prevalence

text: of

text: exozodiacal

text: dust,

text: which

text: were

text: both

text: unknown

text: at

text: that

text: time

text: (e.g.,

text: Defr`ere

text: et

text: al

text: Today,

text: the

text: occurrence

text: of

text: HZ

text: rocky

text: exoplanets

text: has

text: been

text: measured

text: by

text: Kepler

text: (e.g.,

text: Winn

text: and

text: Fabrycky

text: and

text: the

text: prevalence

text: of

text: exozodiacal

text: dust

text: has

text: been

text: very

text: well

text: constrained

text: by

text: ground-based

text: nulling

text: interferometers

text: (Mennesson

text: et

text: al

text: Ertel

text: et

text: al.

text: in

text: prep).

text: It

text: is

text: therefore

text: possible

text: to

text: predict

text: the

text: exoplanet

text: yield

text: of

text: direct

text: imaging

text: instruments

text: more

text: precisely.

text: Recently,

text: based

text: on

text: planet

text: occur-

text: rence

text: statistics

text: from

text: Kepler

text: and

text: using

text: Monte-Carlo

text: simulations,

text: Kammerer

text: and

text: Quanz

text: estimated

text: that

text: exoplanets

text: with

text: radii

text: between

text: R⊕

text: and

text: R⊕

text: can

text: be

text: detected

text: during

text: years

text: of

text: mission

text: time

text: assuming

text: four

text: apertures

text: and

text: throughputs

text: times

text: worse

text: than

text: those

text: for

text: the

text: JWST

text: and

text: overheads.

text: Approximately

text: planets

text: could

text: be

text: habitable

text: (radii

text: be-

text: tween

text: R⊕

text: and

text: R⊕

text: and

text: equilibrium

text: temperatures

text: between

text: K

text: and

text: K)

text: and

text: would

text: be

text: prime

text: targets

text: for

text: follow-up

text: spectroscopic

text: observations.

text: While

text: this

text: exoplanet

text: yield

text: can

text: be

text: interpreted

text: as

text: an

text: upper

text: limit

text: because

text: no

text: ex-

text: ozodiacal

text: dust

text: was

text: assumed

text: in

text: this

text: study,

text: new

text: results

text: from

text: the

text: NASA’s

text: Hunt

text: for

text: Observable

text: Signatures

text: of

text: Terrestrial

text: planetary

text: Systems

text: (HOSTS,

text: Danchi

text: et

text: al

text: survey

text: on

text: the

text: Large

text: Binocular

text: Telescope

text: Interferometer

text: (HOSTS,

text: Hinz

text: et

text: al

text: suggest

text: that

text: exozodiacal

text: dust

text: disks

text: would

text: not

text: be

text: a

text: major

text: ##LTLine##


text: 10 Fig. 2 The left panel shows the planet signal detected with the Planet Detection Testbed (Martin et al 2012). Each point is an item of data from the 2-s chop cycle and the whole trace shows the null signal obtained over a 360 degree eﬀective rotation of the interferometer array. × − The line is a ﬁt to the signal from a planet at a nominal angular radius of 6.35 10 7 rad (or 132mas) from the star. By comparison, the equivalent angular fringe distance from the short × − baseline is 4.7 10 7 rad. Near the center and at the ends of the plot, the planet crosses the null fringe. The right panel shows the equivalent sensitivity map of the interferometer array. Array rotation causes the planet location to orbit (solid line) around the central null fringe (gray), and thus its signal is modulated both by the higher frequency fringes on the long baseline and by the chopping. (Martin et al 2012, see Figure 2), but with ﬂuxes much higher than those ex- pected from stars and planets allowing working at room temperature without being disturbed by the thermal emission of the environment. In parallel, the operation of high-precision ground-based interferometers has matured in both Europe and the US. In particular, Europe has gained a strong expertise in the ﬁeld of fringe sensing, tracking, and stabilization with the operation of the Very Large Telescope Interferometer (VLTI). In the United states, consider- able technical expertise was gained by operating several nulling interferometers such as the Keck Interferometer Nuller (KIN, Colavita et al 2009), the Palomar Fiber Nuller (PFN, Mennesson et al 2011b), and the Large Binocular Tele- scope Interferometer (Hinz et al 2016). All have produced excellent scientiﬁc results (e.g., Mennesson et al 2014; Defr`ere et al 2015), which have pushed high-resolution mid-infrared imaging to new limits (Defr`ere et al 2016). New innovative data reduction techniques have also been developed to improve the accuracy of nulling instruments (Hanot et al 2011; Mennesson et al 2011a) but more work is required to adapt this technique to four-telescope conﬁgurations. 4 Prospects 4.1 Current context Because it is a very challenging observational task partially requiring fur- ther technological developments, the spectroscopic characterisation of small HZ exoplanets by direct imaging is currently not possible with existing in- struments. Future instruments considered to achieve this goal cover generally ##LTFigure## ##LTFigure## ##LTLine##

text: Fig.

text: The

label: caption

text: left

text: panel

text: shows

text: the

text: planet

text: signal

text: detected

text: with

text: the

text: Planet

text: Detection

text: Testbed

text: (Martin

text: et

text: al

text: Each

text: point

text: is

text: an

text: item

text: of

text: data

text: from

text: the

text: chop

text: cycle

text: and

text: the

text: whole

text: trace

text: shows

text: the

text: signal

text: obtained

text: over

text: a

text: degree

text: eﬀective

text: rotation

text: of

text: the

text: interferometer

text: array.

text: The

text: line

text: is

text: a

text: ﬁt

text: to

text: the

text: signal

text: from

text: a

text: planet

text: at

text: a

text: nominal

text: angular

text: radius

text: of

text: rad

text: (or

text: from

text: the

text: star.

text: By

text: comparison,

text: the

text: equivalent

text: angular

text: fringe

text: distance

text: from

text: the

text: short

text: baseline

text: is

text: rad.

text: Near

text: the

text: center

text: and

text: at

text: the

text: ends

text: of

text: the

text: plot,

text: the

text: planet

text: crosses

text: the

text: fringe.

text: The

text: right

text: panel

text: shows

text: the

text: equivalent

text: sensitivity

text: map

text: of

text: the

text: interferometer

text: array.

text: Array

text: rotation

text: causes

text: the

text: planet

text: location

text: to

text: orbit

text: (solid

text: line)

text: around

text: the

text: central

text: fringe

text: (gray),

text: and

text: thus

text: its

text: signal

text: is

text: modulated

text: both

text: by

text: the

text: higher

text: frequency

text: fringes

text: on

text: the

text: long

text: baseline

text: and

text: by

text: the

text: chopping.

text: (Martin

text: et

text: al

text: see

text: Figure

text: but

text: with

text: ﬂuxes

text: much

text: higher

text: than

text: those

text: ex-

text: pected

text: from

text: stars

text: and

text: planets

text: allowing

text: working

text: at

text: room

text: temperature

text: without

text: being

text: disturbed

text: by

text: the

text: thermal

text: emission

text: of

text: the

text: environment.

text: In

text: parallel,

text: the

text: operation

text: of

text: high-precision

text: ground-based

text: interferometers

text: has

text: matured

text: in

text: both

text: Europe

text: and

text: the

text: US.

text: In

text: particular,

text: Europe

text: has

text: gained

text: a

text: strong

text: expertise

text: in

text: the

text: ﬁeld

text: of

text: fringe

text: sensing,

text: tracking,

text: and

text: stabilization

text: with

text: the

text: operation

text: of

text: the

text: Very

text: Large

text: Telescope

text: Interferometer

text: (VLTI).

text: In

text: the

text: United

text: states,

text: consider-

text: able

text: technical

text: expertise

text: was

text: gained

text: by

text: operating

text: several

text: nulling

text: interferometers

text: such

text: as

text: the

text: Keck

text: Interferometer

text: Nuller

text: (KIN,

text: Colavita

text: et

text: al

text: the

text: Palomar

text: Fiber

text: Nuller

text: (PFN,

text: Mennesson

text: et

text: al

text: and

text: the

text: Large

text: Binocular

text: Tele-

text: scope

text: Interferometer

text: (Hinz

text: et

text: al

text: All

text: have

text: produced

text: excellent

text: scientiﬁc

text: results

text: (e.g.,

text: Mennesson

text: et

text: al

text: Defr`ere

text: et

text: al

text: which

text: have

text: pushed

text: high-resolution

text: mid-infrared

text: imaging

text: to

text: new

text: limits

text: (Defr`ere

text: et

text: al

text: New

text: innovative

text: data

text: reduction

text: techniques

text: have

text: also

text: been

text: developed

text: to

text: improve

text: the

text: accuracy

text: of

text: nulling

text: instruments

text: (Hanot

text: et

text: al

text: Mennesson

text: et

text: al

text: but

text: more

text: work

text: is

text: required

text: to

text: adapt

text: this

text: technique

text: to

text: four-telescope

text: conﬁgurations.

text: Prospects

text: Current

text: context

text: Because

text: it

text: is

text: a

text: very

text: challenging

text: observational

text: task

text: partially

text: requiring

text: fur-

text: ther

text: technological

text: developments,

text: the

text: spectroscopic

text: characterisation

text: of

text: small

text: HZ

text: exoplanets

text: by

text: direct

text: imaging

text: is

text: currently

text: not

text: possible

text: with

text: existing

text: in-

text: struments.

text: Future

text: instruments

text: considered

text: to

text: achieve

text: this

text: goal

text: cover

text: generally

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##


text: Finally, we believe that the methodology has the potential to impact a wide range of applications domains, the most obvious being areas where co-kriging/multi- output GPs are already being used. Other areas also include multivariate time series analysis and time-evolving computer models Conti and O’Hagan [2010], model comparison in Bayesian statistics or even the development of new proba- bilistic numerical methods. Acknowledgements FXB was supported by the EPSRC grants [EP/L016710/1, EP/R018413/1]. MG was supported by the EPSRC grants [EP/J016934/3, EP/K034154/1, EP/P020720/1, EP/R018413/1], an EPSRC Established Career Fellowship, the EU grant [EU/259348] and the Lloyds Register Foundation Programme on Data-Centric Engineering. This work was also supported by The Alan Turing Institute under the EPSRC grant [EP/N510129/1]. Finally, this material was also based upon work partially supported by the National Science Foundation under Grant DMS-1127914 to the Statistical and Applied Mathematical Sciences Institute. Any opinions, ﬁndings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reﬂect the views of the National Science Foun- dation. References M. A. A´lvarez and N. D. Lawrence. Computationally Eﬃcient Convolved Multiple Output Gaussian Processes. Journal of Machine Learning Research, 12:1459– 1500, 2011. M. A Alvarez, L. Rosasco, and N. D. Lawrence. Kernels for Vector-Valued Func- tions: A Review. Foundations and Trends in Machine Learning, 4(3):195–266, 2012. F Bach. On the Equivalence between Quadrature Rules and Random Features. Journal of Machine Learning Research, 18(19), 2017. N. S. Bakhvalov. On the optimality of linear methods for operator approximation in convex classes of functions. USSR Computational Mathematics and Mathe- matical Physics, 11(4):244–249, 1971. A. Berlinet and C. Thomas-Agnan. Reproducing Kernel Hilbert Spaces in Proba- bility and Statistics. Springer Science+Business Media, New York, 2004. F.-X. Briol, C. J. Oates, M. Girolami, and M. A. Osborne. Frank-Wolfe Bayesian Quadrature: Probabilistic Integration with Theoretical Guarantees. In Ad- vances In Neural Information Processing Systems 28, pages 1162—-1170, 2015a. F-X. Briol, C. J. Oates, M. Girolami, M. A. Osborne, and D. Sejdinovic. Probabilistic Integration: A Role for Statisticians in Numerical Analysis? arXiv:1512.00933, pages 1–63, 2015b. F-X. Briol, C. J. Oates, J. Cockayne, and M. Girolami. On the Sampling Problem for Kernel Quadrature. In Proceedings of the 34th International Conference on Machine Learning, PMLR 70, pages 586–595, 2017. J. Brouillat, C. Bouville, B. Loos, C. Hansen, and K. Bouatouch. A Bayesian Monte Carlo Approach to global illumination. Computer Graphics Forum, 28 (8):2315–2329, 2009. A Caponnetto and E De Vito. Optimal rates for the regularized least-squares algorithm. Foundations of Computational Mathematics, pages 331–368, 2007. 14

text: Finally,

text: we

text: believe

text: that

text: the

text: methodology

text: has

text: the

text: potential

text: to

text: impact

text: a

text: wide

text: range

text: of

text: applications

text: domains,

text: the

text: most

text: obvious

text: being

text: areas

text: where

text: co-kriging/multi-

text: output

text: GPs

text: are

text: already

text: being

text: used.

text: Other

text: areas

text: also

text: include

text: multivariate

text: time

text: series

text: analysis

text: and

text: time-evolving

text: computer

text: models

text: Conti

text: and

text: O’Hagan

text: model

text: comparison

text: in

text: Bayesian

text: statistics

text: or

text: even

text: the

text: development

text: of

text: new

text: proba-

text: bilistic

text: numerical

text: methods.

text: Acknowledgements

label: section

text: FXB

text: was

text: supported

text: by

text: the

text: EPSRC

text: grants

text: MG

text: was

text: supported

text: by

text: the

text: EPSRC

text: grants

text: an

text: EPSRC

text: Established

text: Career

text: Fellowship,

text: the

text: EU

text: grant

text: and

text: the

text: Lloyds

text: Register

text: Foundation

text: Programme

text: on

text: Data-Centric

text: Engineering.

text: This

text: work

text: was

text: also

text: supported

text: by

text: The

text: Alan

text: Turing

text: Institute

text: under

text: the

text: EPSRC

text: grant

text: Finally,

text: this

text: material

text: was

text: also

text: based

text: upon

text: work

text: partially

text: supported

text: by

text: the

text: National

text: Science

text: Foundation

text: under

text: Grant

text: to

text: the

text: Statistical

text: and

text: Applied

text: Mathematical

text: Sciences

text: Institute.

text: Any

text: opinions,

text: ﬁndings,

text: and

text: conclusions

text: or

text: recommendations

text: expressed

text: in

text: this

text: material

text: are

text: those

text: of

text: the

text: author(s)

text: and

text: do

text: not

text: necessarily

text: reﬂect

text: the

text: views

text: of

text: the

text: National

text: Science

text: Foun-

text: dation.

text: References

text: M.

text: A.

text: A´lvarez

text: and

text: N.

text: D.

text: Lawrence.

text: Computationally

text: Eﬃcient

text: Convolved

text: Multiple

text: Output

text: Gaussian

text: Processes.

text: Journal

text: of

text: Machine

text: Learning

text: Research,

text: M.

text: A

text: Alvarez,

text: L.

text: Rosasco,

text: and

text: N.

text: D.

text: Lawrence.

text: Kernels

text: for

text: Vector-Valued

text: Func-

text: tions:

text: A

text: Review.

text: Foundations

text: and

text: Trends

text: in

text: Machine

text: Learning,

text: F

text: Bach.

text: On

text: the

text: Equivalence

text: between

text: Quadrature

text: Rules

text: and

text: Random

text: Features.

text: Journal

text: of

text: Machine

text: Learning

text: Research,

text: N.

text: S.

text: Bakhvalov.

text: On

text: the

text: optimality

text: of

text: linear

text: methods

text: for

text: operator

text: approximation

text: in

text: convex

text: classes

text: of

text: functions.

text: USSR

text: Computational

text: Mathematics

text: and

text: Mathe-

text: matical

text: Physics,

text: A.

text: Berlinet

text: and

text: C.

text: Thomas-Agnan.

text: Reproducing

text: Kernel

text: Hilbert

text: Spaces

text: in

text: Proba-

text: bility

text: and

text: Statistics.

text: Springer

text: Science+Business

text: Media,

text: New

text: York,

text: F.-X.

text: Briol,

text: C.

text: J.

text: Oates,

text: M.

text: Girolami,

text: and

text: M.

text: A.

text: Osborne.

text: Frank-Wolfe

text: Bayesian

text: Quadrature:

text: Probabilistic

text: Integration

text: with

text: Theoretical

text: Guarantees.

text: In

text: Ad-

text: vances

text: In

text: Neural

text: Information

text: Processing

text: Systems

text: pages

text: F-X.

text: Briol,

text: C.

text: J.

text: Oates,

text: M.

text: Girolami,

text: M.

text: A.

text: Osborne,

text: and

text: D.

text: Sejdinovic.

text: Probabilistic

text: Integration:

text: A

text: Role

text: for

text: Statisticians

text: in

text: Numerical

text: Analysis?

text: pages

text: F-X.

text: Briol,

text: C.

text: J.

text: Oates,

text: J.

text: Cockayne,

text: and

text: M.

text: Girolami.

text: On

text: the

text: Sampling

text: Problem

text: for

text: Kernel

text: Quadrature.

text: In

text: Proceedings

text: of

text: the

text: International

text: Conference

text: on

text: Machine

text: Learning,

text: PMLR

text: pages

text: J.

text: Brouillat,

text: C.

text: Bouville,

text: B.

text: Loos,

text: C.

text: Hansen,

text: and

text: K.

text: Bouatouch.

text: A

text: Bayesian

text: Monte

text: Carlo

text: Approach

text: to

text: global

text: illumination.

text: Computer

text: Graphics

text: Forum,

text: A

text: Caponnetto

text: and

text: E

text: De

text: Vito.

text: Optimal

text: rates

text: for

text: the

text: regularized

text: least-squares

text: algorithm.

text: Foundations

text: of

text: Computational

text: Mathematics,

text: pages


text: for some constants σ , λ > 0 for i = 1, . . . , 6. Note that for simplicity, we did i i not include c and c . The kernel mean and initial error can easily be computed w1 w2 in closed form using Gaussian identities. B.2 Hyper-parameters One of the main challenges when using uni-output BQ and multi-output BQ is the selection of appropriate hyperparameters. In this section, we consider multi- GP output BQ with (0, C) prior we denote the hyperparameters of the kernel C in vector form as θ = (θ , . . . , θ ). To optimise these parameters, we propose to 1 l use an empirical-Bayes approach and maximise the log-marginal likelihood: | −1 (cid:62) − log p (f(X) X, θ) = f(X) C(X, X) 1f(X) 2 −1 | | − ND log C(X, X) log(2π). 2 2 This can be eﬃciently optimised by making use of gradients of the log-marginal ∀ ∈ { } likelihood i 1, . . . , l : | (cid:18) (cid:19) ∂ log p (f(X) X, θ) 1 (cid:62) − ∂C(X, X) − = f(X) C(X, X) 1 C(X, X) 1f(X) ∂θ 2 ∂θ i i −1 − ∂C(X, X) Tr C(X, X) 1 2 ∂θ i C Extended Numerical Experiments In this appendix, we provide additional results for the multi-output BQ experi- ments provided in Sec. for the multiﬁdelity toy models and the global illumination problem. C.1 Multi-ﬁdelity modelling In Fig. 5, we give an extended version of Fig. 1 which includes credible intervals for each of the multi-output GP models. 21 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: for

text: some

text: constants

text: σ

text: λ

text: for

text: i

text: Note

text: that

text: for

text: simplicity,

text: we

text: did

text: i

text: i

text: not

text: include

text: c

text: and

text: c

text: The

text: kernel

text: mean

text: and

text: initial

text: error

text: can

text: easily

text: be

text: computed

text: in

text: closed

text: form

text: using

text: Gaussian

text: identities.

text: Hyper-parameters

text: One

text: of

text: the

text: main

text: challenges

text: when

text: using

text: uni-output

text: BQ

text: and

text: multi-output

text: BQ

text: is

text: the

text: selection

text: of

text: appropriate

text: hyperparameters.

text: In

text: this

text: section,

text: we

text: consider

text: multi-

text: GP

text: output

text: BQ

text: with

text: C)

text: prior

text: we

text: denote

text: the

text: hyperparameters

text: of

text: the

text: kernel

text: C

text: in

text: vector

text: form

text: as

text: θ

text: (θ

text: θ

text: To

text: optimise

text: these

text: parameters,

text: we

text: propose

text: to

text: l

text: use

text: an

text: empirical-Bayes

text: approach

text: and

text: maximise

text: the

text: log-marginal

text: likelihood:

text: −1

text: log

text: p

text: (f(X)

text: X,

text: θ)

text: f(X)

text: C(X,

text: X)

text: −1

text: ND

text: log

text: C(X,

text: X)

text: This

text: can

text: be

text: eﬃciently

text: optimised

text: by

text: making

text: use

text: of

text: gradients

text: of

text: the

text: log-marginal

text: likelihood

text: i

text: l

text: log

text: p

text: (f(X)

text: X,

text: θ)

text: ∂C(X,

text: X)

text: f(X)

text: C(X,

text: X)

text: C(X,

text: X)

text: ∂θ

text: ∂θ

text: i

text: i

text: −1

text: ∂C(X,

text: X)

text: Tr

text: C(X,

text: X)

text: ∂θ

text: i

text: C

text: Extended

text: Numerical

text: Experiments

text: In

text: this

text: appendix,

text: we

text: provide

text: additional

text: results

text: for

text: the

text: multi-output

text: BQ

text: experi-

text: ments

text: provided

text: in

text: Sec.

text: for

text: the

text: multiﬁdelity

text: toy

text: models

text: and

text: the

text: global

text: illumination

text: problem.

text: Multi-ﬁdelity

text: modelling

text: In

text: Fig.

text: we

text: give

text: an

text: extended

text: version

text: of

text: Fig.

text: which

text: includes

text: credible

text: intervals

text: for

text: each

text: of

text: the

text: multi-output

text: GP

text: models.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Figure 5: Multi-ﬁdelity modelling: Plot of the Step function (top) and Forrester func- tion (bottom) with Gaussian Process credible intervals. C.2 Global illumination problem In Fig. 6, we plot the evolution of the worst-case integration error as N increases for the uni-output, two-output and ﬁve-output BQ with LMC kernel. As expected from Thm 1, the convergence occurs at the same rate in N but with a smaller rate constant the more outputs there are. 22 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure##

text: Figure

text: Multi-ﬁdelity

text: modelling:

text: Plot

text: of

text: the

text: Step

text: function

text: (top)

text: and

text: Forrester

text: func-

text: tion

text: (bottom)

text: with

text: Gaussian

text: Process

text: credible

text: intervals.

text: Global

text: illumination

text: problem

text: In

text: Fig.

text: we

text: plot

text: the

text: evolution

text: of

text: the

text: worst-case

text: integration

text: error

text: as

text: N

text: increases

text: for

text: the

text: uni-output,

text: two-output

text: and

text: ﬁve-output

text: BQ

text: with

text: LMC

text: kernel.

text: As

text: expected

text: from

text: Thm

text: the

text: convergence

text: occurs

text: at

text: the

text: same

text: rate

text: in

text: N

text: but

text: with

text: a

text: smaller

text: rate

text: constant

text: the

text: more

text: outputs

text: there

text: are.

text: ##LTFigure##

label: figure

font: default

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##


text: 16 COLETTE DE COSTER AND ANTONIO J. FERNA´NDEZ (cid:90) (cid:104) (cid:0) (cid:1) (cid:105) On the other hand, by deﬁnition of g, observe that λc(x)g(v)+ 1 +(cid:90) (cid:104)−µ v (cid:12)(cid:12)p−1h(x) (cid:0)ϕ dx (cid:1)(cid:12)(cid:12) (cid:16) (cid:0) (cid:1)(cid:17) (cid:105) p 1 (cid:12) (cid:12) Ω (cid:90) − − − (cid:2) p 1 (cid:3) µ p 2 p 1 µ = λc(x) ln 1 + − v ln 1 + − v + h(x) φ dx µ p 1 µ p 1 Ω | | − = λc(x) u p 2u + h(x) φ dx. (cid:90) Ω (cid:90) (cid:2) (cid:3) As v is a solution of (Q ) we deduce from these two identities that λ |∇ | − ∇ ∇ | | − |∇ | u p 2 u φ dx = λc(x) u p 2u + µ u p + h(x) φ dx, Ω Ω and so, u is a solution of (Pλ), as desired. (cid:0) (cid:1) ∈ ∩ ∞ In the same way, assume that u W1,p(Ω) L (Ω) is a solution of (P ). By Proposition 4.2 we know that u ≥ uλ. Hence, it follows that v = p0−µ1 epµ−u1 − 1 ∈ W01,p(Ω) ∩ L∞(Ω) λand satisﬁes v ≥ αλ ≥ −p−µ1 +(cid:3)ε for some ε > 0. Arguing exactly as before, we deduce that v is a solution of (Q ). λ ∈ ∩ ∞ Remark 5.1. Arguing exactly as in the proof of Lemma 5.2, iii), we can show that v H1(Ω) L (Ω) ∈ ∩ ∞ 1 (respectively v2 H1(Ω) L(cid:0) (Ω)) is a l(cid:1)ower solut(cid:16)ion (respectively an upper s(cid:0)olution) of ((cid:1)Q(cid:17)λ) if, and only if, the function − − p 1 µ p 1 µ u = ln 1 + − v respectively u = ln 1 + − v 1 µ p 1 1 2 µ p 1 2 is a lower solution (respectively an upper solution) of (P ). λ The interest of problem (Q ) comes from the fact that it has a variational formulation. We can obtain the solutions of (Q ) as criticaλl points of the(cid:90)functional I (cid:90): W1,p(Ω) → R deﬁned as λ λ 0 1 |∇ | − (5.5) (cid:82) I (v) = v p dx F (x, v)dx , λ p λ Ω Ω where we deﬁne F (x, s) = s f (x, t) dt i.e. (cid:16) (cid:17) λ 0 λ − p 1 µ p ≥ (5.6) F (x, s) = λc(x)G(s) + 1 + − s h(x), if s α (x) , λ µp p 1 λ and (cid:104) (cid:0) (cid:1) (cid:105) µ p−1 (cid:0) − (cid:1) F (x, s) = λc(x)g(α (x)) + 1 + − α (x) h(x) (s α ) λ λ p 1 λ λ (5.7) − p 1 µ ≤ p + λc(x)G(α (x)) + 1 + − α (x) h(x), if s α (x) . λ µp p 1 λ λ Observe that under the assumptions (A ), since g has subcritical growth (see Lemma 5.1), I ∈ C1(W1,p(Ω), R) 1 0 (see for example [22] page 356). ∈ R Lemma 5.3. Assume that (A ) holds and let λ be arbitrary, Then, any bounded Cerami sequence for 1 I admits a convergent subsequence. λ Proof. Let {v } ⊂ W1,p(Ω) be a bounded Cerami sequence for I at level d ∈ R. We are going to show that, n 0 → ∈ ∈ λ up to a subsequence, v v W1,p(Ω) for a v W1,p(Ω). n 0 0 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: COLETTE

text: DE

text: COSTER

text: AND

text: ANTONIO

text: J.

text: FERNA´NDEZ

text: On

text: the

text: other

text: hand,

text: by

text: deﬁnition

text: of

text: g,

text: observe

text: that

text: λc(x)g(v)+

text: v

text: dx

text: p

text: Ω

text: p

text: µ

text: p

text: p

text: µ

text: λc(x)

text: ln

text: v

text: ln

text: v

text: h(x)

text: φ

text: dx

text: µ

text: p

text: µ

text: p

text: Ω

text: λc(x)

text: u

text: p

text: h(x)

text: φ

text: dx.

text: Ω

text: As

text: v

text: is

text: a

text: solution

text: of

text: (Q

text: we

text: deduce

text: from

text: these

text: two

text: identities

text: that

text: λ

text: u

text: p

text: u

text: φ

text: dx

text: λc(x)

text: u

text: p

text: µ

text: u

text: p

text: h(x)

text: φ

text: dx,

text: Ω

text: Ω

text: and

text: so,

text: u

text: is

text: a

text: solution

text: of

text: (Pλ),

text: as

text: desired.

text: In

text: the

text: same

text: way,

text: assume

text: that

text: u

text: L

text: (Ω)

text: is

text: a

text: solution

text: of

text: (P

text: By

text: Proposition

text: we

text: know

text: that

text: u

text: uλ.

text: Hence,

text: it

text: follows

text: that

text: v

text: L∞(Ω)

text: λand

text: satisﬁes

text: v

text: αλ

text: −p−µ1

text: for

text: some

text: ε

text: Arguing

text: exactly

text: as

text: before,

text: we

text: deduce

text: that

text: v

text: is

text: a

text: solution

text: of

text: (Q

text: λ

text: Remark

text: Arguing

text: exactly

text: as

text: in

text: the

text: proof

text: of

text: Lemma

text: iii),

text: we

text: can

text: show

text: that

text: v

text: L

text: (Ω)

text: (respectively

text: (Ω))

text: is

text: a

text: (respectively

text: an

text: upper

text: of

text: if,

text: and

text: only

text: if,

text: the

text: function

text: p

text: µ

text: p

text: µ

text: u

text: ln

text: v

text: respectively

text: u

text: ln

text: v

text: µ

text: p

text: µ

text: p

text: is

text: a

text: lower

text: solution

text: (respectively

text: an

text: upper

text: solution)

text: of

text: (P

text: λ

text: The

text: interest

text: of

text: problem

text: (Q

text: comes

text: from

text: the

text: fact

text: that

text: it

text: has

text: a

text: variational

text: formulation.

text: We

text: can

text: obtain

text: the

text: solutions

text: of

text: (Q

text: as

text: criticaλl

text: points

text: of

text: I

text: R

text: deﬁned

text: as

text: λ

text: λ

text: I

text: (v)

text: v

text: p

text: dx

text: F

text: (x,

text: v)dx

text: λ

text: p

text: λ

text: Ω

text: Ω

text: where

text: we

text: deﬁne

text: F

text: (x,

text: s)

text: s

text: f

text: (x,

text: t)

text: dt

text: i.e.

text: λ

text: λ

text: p

text: µ

text: p

text: F

text: (x,

text: s)

text: λc(x)G(s)

text: s

text: h(x),

text: if

text: s

text: α

text: (x)

text: λ

text: µp

text: p

text: λ

text: and

text: µ

text: F

text: (x,

text: s)

text: λc(x)g(α

text: (x))

text: α

text: (x)

text: h(x)

text: (s

text: α

text: λ

text: λ

text: p

text: λ

text: λ

text: p

text: µ

text: p

text: λc(x)G(α

text: (x))

text: α

text: (x)

text: h(x),

text: if

text: s

text: α

text: (x)

text: λ

text: µp

text: p

text: λ

text: λ

text: Observe

text: that

text: under

text: the

text: assumptions

text: (A

text: since

text: g

text: has

text: subcritical

text: growth

text: (see

text: Lemma

text: I

text: R)

text: (see

text: for

text: example

text: page

text: R

text: Lemma

text: Assume

text: that

text: (A

text: holds

text: and

text: let

text: λ

text: be

text: arbitrary,

text: Then,

text: any

text: bounded

text: Cerami

text: sequence

text: for

text: I

text: admits

text: a

text: convergent

text: subsequence.

text: λ

text: Proof.

text: Let

text: {v

text: be

text: a

text: bounded

text: Cerami

text: sequence

text: for

text: I

text: at

text: level

text: d

text: R.

text: We

text: are

text: going

text: to

text: show

text: that,

text: n

text: λ

text: up

text: to

text: a

text: subsequence,

text: v

text: v

text: for

text: a

text: v

text: n

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: A P-LAPLACIAN PROBLEM WITH CRITICAL GROWTH IN THE GRADIENT 3 We can state the following result. Theorem 1.3. Assume that (A ) holds. Then (P ) has a solution if, and only if, m > 0. 1 0 p Theorem 1.3 provides, so to say, a characterization in term of a ﬁrst eigenvalue of the existence of solution (cid:8) to (P ). This result again improves, for µ constant, [8] and it allows to observe that, in case h 0, (P ) has 0 (cid:9) 0 always a solution while the case h 0 is the “worse” case for the existence of a solution. In case h changes sign, the negative part of h “helps” in order to have a solution to (P ). We give in Appendix , suﬃcient 0 conditions on h+ in order to ensure m > 0. p Remark 1.1. Observe that the suﬃcient part of Theorem 1.3 is direct. Indeed, if m > 0 then m+ > 0 and − p p,0 m > 0 and Theorem 1.1 implies that (P ) has a solution. p,0 0 Remark 1.2. We see, combining Theorems 1.1 and 1.3, that if (P ) has a solution then (P ) has a solution ≤ 0 λ for any λ 0. Moreover this solution is unique by Theorem 1.2. Now, we turn to the study the non-coercive case, namely when λ > 0. First, using mainly variational techniques we prove the following result. Theorem 1.4. Assume that (A ) holds and suppose that (P ) has a solution. Then there exists Λ > 0 such 1 0 that, for any 0 < λ < Λ, (P ) has at least two solutions. λ As we shall see in Corollary 9.4, the existence of a solution to (P ) is, in some sense, necessary for the 0 existence of a solution when λ > 0. Next, considering stronger regularity assumptions, we derive informations on the structure of the set of solutions in the non-coercive case. These informations complement Theorem 1.4. We denote by γ > 0 the 1 ﬁrst eigenvalue of the problem − | | − ∈ (1.2) ∆ u = γc(x) u p 2u , u W1,p(Ω) , p 0  and, under the assumptions ⊂ R ≥ C Ω N, N 2 , is a bounded domain with ∂Ω of class 2, ∞ (A ) c and h belong to L (Ω) , 2 (cid:9) c 0 and µ > 0 , we state the following theorem. Theorem 1.5. Assume that (A ) holds and suppose that (P ) has a solution. Then: • (cid:8) 2 0 (cid:28) If h 0, for every λ > 0, (P ) has at least two solutions u , u with u 0. • (cid:9) (cid:29) λ ∈ 1 2 1 If h 0, then u 0 and there exists λ (0, γ ) such that: 0 1 ≥ – for every 0 < λ < λ, (P ) has at least two solutions satisfying u u ; λ ≥ i 0 – for λ = λ, (P ) has at least one solution satisfying u u ; λ 0 – for any λ > λ, (P ) has no non-negative solution. λ ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: A

text: P-LAPLACIAN

text: PROBLEM

text: WITH

text: CRITICAL

text: GROWTH

text: IN

text: THE

text: GRADIENT

text: We

text: can

text: state

text: the

text: following

text: result.

text: Theorem

text: Assume

text: that

text: (A

text: holds.

text: Then

text: (P

text: has

text: a

text: solution

text: if,

text: and

text: only

text: if,

text: m

text: p

text: Theorem

text: provides,

text: so

text: to

text: say,

text: a

text: characterization

text: in

text: term

text: of

text: a

text: ﬁrst

text: eigenvalue

text: of

text: the

text: existence

text: of

text: solution

text: to

text: (P

text: This

text: result

text: again

text: improves,

text: for

text: µ

text: constant,

text: and

text: it

text: allows

text: to

text: observe

text: that,

text: in

text: case

text: h

text: (P

text: has

text: always

text: a

text: solution

text: while

text: the

text: case

text: h

text: is

text: the

text: “worse”

text: case

text: for

text: the

text: existence

text: of

text: a

text: solution.

text: In

text: case

text: h

text: changes

text: sign,

text: the

text: negative

text: part

text: of

text: h

text: “helps”

text: in

text: order

text: to

text: have

text: a

text: solution

text: to

text: (P

text: We

text: give

text: in

text: Appendix

text: suﬃcient

text: conditions

text: on

text: h+

text: in

text: order

text: to

text: ensure

text: m

text: p

text: Remark

text: Observe

text: that

text: the

text: suﬃcient

text: part

text: of

text: Theorem

text: is

text: direct.

text: Indeed,

text: if

text: m

text: then

text: m+

text: and

text: p

text: m

text: and

text: Theorem

text: implies

text: that

text: (P

text: has

text: a

text: solution.

text: Remark

text: We

text: see,

text: combining

text: Theorems

text: and

text: that

text: if

text: (P

text: has

text: a

text: solution

text: then

text: (P

text: has

text: a

text: solution

text: λ

text: for

text: any

text: λ

text: Moreover

text: this

text: solution

text: is

text: unique

text: by

text: Theorem

text: Now,

text: we

text: turn

text: to

text: the

text: study

text: the

text: non-coercive

text: case,

text: namely

text: when

text: λ

text: First,

text: using

text: mainly

text: variational

text: techniques

text: we

text: prove

text: the

text: following

text: result.

text: Theorem

text: Assume

text: that

text: (A

text: holds

text: and

text: suppose

text: that

text: (P

text: has

text: a

text: solution.

text: Then

text: there

text: exists

text: Λ

text: such

text: that,

text: for

text: any

text: λ

text: Λ,

text: (P

text: has

text: at

text: least

text: two

text: solutions.

text: λ

text: As

text: we

text: shall

text: see

text: in

text: Corollary

text: the

text: existence

text: of

text: a

text: solution

text: to

text: (P

text: is,

text: in

text: some

text: sense,

text: necessary

text: for

text: the

text: existence

text: of

text: a

text: solution

text: when

text: λ

text: Next,

text: considering

text: stronger

text: regularity

text: assumptions,

text: we

text: derive

text: informations

text: on

text: the

text: structure

text: of

text: the

text: set

text: of

text: solutions

text: in

text: the

text: non-coercive

text: case.

text: These

text: informations

text: complement

text: Theorem

text: We

text: denote

text: by

text: γ

text: the

text: ﬁrst

text: eigenvalue

text: of

text: the

text: problem

text: u

text: γc(x)

text: u

text: p

text: u

text: p

text: and,

text: under

text: the

text: assumptions

text: R

text: C

text: Ω

text: N,

text: N

text: is

text: a

text: bounded

text: domain

text: with

text: ∂Ω

text: of

text: class

text: (A

text: c

text: and

text: h

text: belong

text: to

text: L

text: (Ω)

text: c

text: and

text: µ

text: we

text: state

text: the

text: following

text: theorem.

text: Theorem

text: Assume

text: that

text: (A

text: holds

text: and

text: suppose

text: that

text: (P

text: has

text: a

text: solution.

text: Then:

text: If

text: h

text: for

text: every

text: λ

text: (P

text: has

text: at

text: least

text: two

text: solutions

text: u

text: u

text: with

text: u

text: λ

text: If

text: h

text: then

text: u

text: and

text: there

text: exists

text: λ

text: γ

text: such

text: that:

text: for

text: every

text: λ

text: λ,

text: (P

text: has

text: at

text: least

text: two

text: solutions

text: satisfying

text: u

text: u

text: λ

text: i

text: for

text: λ

text: λ,

text: (P

text: has

text: at

text: least

text: one

text: solution

text: satisfying

text: u

text: u

text: λ

text: for

text: any

text: λ

text: λ,

text: (P

text: has

text: no

text: non-negative

text: solution.

text: λ

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: A P-LAPLACIAN PROBLEM WITH CRITICAL GROWTH IN THE GRADIENT 27 (cid:90) (cid:104)(cid:0) (cid:1) (cid:0) (cid:1) (cid:105) (cid:90) (cid:0) (cid:1) As above, we have 1 µ − µ − ≤ − µ − − 1 + − tv p − p(tv)p h (x) dx tp 1 1 + − v p 1h (x) dx. (cid:34)p (cid:90)Ω (cid:16) p (cid:0)1 (cid:1) p 1 (cid:17) (cid:90) Ω p 1 (cid:13)(cid:13) (cid:13)(cid:13) (cid:35) Hence we obtain (cid:13) (cid:13) I (tv) ≤ tp 1 |∇v|p − −µ p−1|v|ph(x) dx − λ c(x)vp G(tv) dx + 1 p − 1 1 + −µ v p−1(cid:107)h−(cid:107) . λ p p 1 tpvp t µ p 1 ∞ 1 Ω (cid:90) Ω Now, since by Lemma 5.1, we have G(tv) ∞ lim λ c(x)vp dx = , →∞ t (tv)p Ω −∞ (cid:3) we deduce that lim I (tv) = from which the lemma follows. →∞ λ t ≥ Proposition 8.4. Assume that (A ) holds and suppose that m > 0. Moreover, suppose that λ 0 is 1 p small enough in order to ensure that the conclusion of Lemma 8.2 holds. Then, I possesses a critical point ∈ ≤ λ v B(0, r) with I (v) I (0), which is a local minimum of I . λ λ λ Proof. From Lemma 8.2, we see that there exists r > 0 such that ≤ (cid:107) (cid:107) m := inf I (v) I (0) and I (v) > I (0) if v = r . ∈ λ λ λ λ v B(0,r) { } ⊂ → { } Let v B(0, r) be such that I (v ) m. Since v is bounded, up to a subsequence, it follows that n∈ λ n n v (cid:42) v W1,p(Ω). By the weak lower semicontinuity of the norm and of the functional I , we have n 0 λ (cid:107) (cid:107) ≤ (cid:107) (cid:107) ≤ ≤ ≤ v lim inf v r and I (v) lim inf I (v ) = m I (0) . →∞ n λ →∞ λ n λ n n (cid:107) (cid:107) ∈ (cid:3) Finally, as I (v) > I (0) if v = r, we deduce that v B(0, r) is a local minimum of I . λ λ λ Proof of Theorem 1.4. Assume that λ > 0 is small enough in order to ensure that the conclusion of Lemma 8.2 holds. By Proposition 8.4 we have a ﬁrst critical point, which is a local minimum of I . On the λ other hand, since the Cerami condition holds, in view of Lemmata 8.2. and 8.3, we can apply Theorem 2.8 and obtain a second critical point of I at the mountain-pass level. This gives two diﬀerent solutions of (Q ). λ λ(cid:3) Finally, by Lemma 5.2, we obtain two solutions of (P ). λ 9. Proof of Theorems 1.5 and 1.6 In this section, we assume the stronger assumption (A ). In that case, we are able to improve our results 2 on the non-coercive case. Proposition 9.1. Assume that (A ) holds with h (cid:8) 0. Then, for every λ > 0, there exists v ∈ C1,τ(Ω), for (cid:28) 2 0 some 0 < τ < 1, with v 0, which is a local minimum of I in the W1,p-topology and a solution of (Q ) ≥ λ 0 λ with v α (with α deﬁned by (5.2)). λ λ (cid:8) Proof. First of all, observe that, as h 0, we have m > 0 and hence, by Theorem 1.3, (P ) has a solution ∈ ∩ ∞ (cid:0) p(cid:1) 0 u W1,p(Ω) L (Ω). By Lemma 5.2, 0 0 − p 1 − ∈ ∩ ∞  v0 = µ(cid:0) ep−µ1u0 (cid:1)1 W01,p(Ω) L (Ω),  is then a weak solution of − µ p−1 (cid:8) ∆ v = 1 + − v h(x) 0, in Ω, p 0 p 1 0 (cid:0) (cid:1) v = 0, on ∂Ω. 0 As moreover, 1+ −µ v p−1h(x) ∈ L∞(Ω), it follows from [21,35] that v ∈ C1,τ(Ω), for some τ ∈ (0, 1) and, p 1 0 (cid:28) 0 0 by the strong maximum principle (see [40]), that v 0. Now, we split the rest of the proof in three steps. 0 Step 1: 0 is a strict upper solution of (Q ). λ ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: A

text: P-LAPLACIAN

text: PROBLEM

text: WITH

text: CRITICAL

text: GROWTH

text: IN

text: THE

text: GRADIENT

text: As

text: above,

text: we

text: have

text: µ

text: µ

text: µ

text: tv

text: p

text: p(tv)p

text: h

text: (x)

text: dx

text: tp

text: v

text: p

text: (x)

text: dx.

text: p

text: p

text: Ω

text: p

text: Hence

text: we

text: obtain

text: I

text: (tv)

text: tp

text: |∇v|p

text: −µ

text: dx

text: λ

text: c(x)vp

text: G(tv)

text: dx

text: p

text: −µ

text: v

text: λ

text: p

text: p

text: tpvp

text: t

text: µ

text: p

text: Ω

text: Ω

text: Now,

text: since

text: by

text: Lemma

text: we

text: have

text: G(tv)

text: lim

text: λ

text: c(x)vp

text: dx

text: t

text: (tv)p

text: Ω

text: we

text: deduce

text: that

text: lim

text: I

text: (tv)

text: from

text: which

text: the

text: lemma

text: follows.

text: λ

text: t

text: Proposition

text: Assume

text: that

text: (A

text: holds

text: and

text: suppose

text: that

text: m

text: Moreover,

text: suppose

text: that

text: λ

text: is

text: p

text: small

text: enough

text: in

text: order

text: to

text: ensure

text: that

text: the

text: conclusion

text: of

text: Lemma

text: holds.

text: Then,

text: I

text: possesses

text: a

text: critical

text: point

text: λ

text: v

text: r)

text: with

text: I

text: (v)

text: I

text: which

text: is

text: a

text: local

text: minimum

text: of

text: I

text: λ

text: λ

text: λ

text: Proof.

text: From

text: Lemma

text: we

text: see

text: that

text: there

text: exists

text: r

text: such

text: that

text: m

text: inf

text: I

text: (v)

text: I

text: and

text: I

text: (v)

text: I

text: if

text: v

text: r

text: λ

text: λ

text: λ

text: λ

text: v

text: Let

text: v

text: r)

text: be

text: such

text: that

text: I

text: (v

text: m.

text: Since

text: v

text: is

text: bounded,

text: up

text: to

text: a

text: subsequence,

text: it

text: follows

text: that

text: n∈

text: λ

text: n

text: n

text: v

text: v

text: By

text: the

text: weak

text: lower

text: semicontinuity

text: of

text: the

text: norm

text: and

text: of

text: the

text: functional

text: I

text: we

text: have

text: n

text: λ

text: v

text: lim

text: inf

text: v

text: r

text: and

text: I

text: (v)

text: lim

text: inf

text: I

text: (v

text: m

text: I

text: n

text: λ

text: λ

text: n

text: λ

text: n

text: n

text: Finally,

text: as

text: I

text: (v)

text: I

text: if

text: v

text: r,

text: we

text: deduce

text: that

text: v

text: r)

text: is

text: a

text: local

text: minimum

text: of

text: I

text: λ

text: λ

text: λ

text: Proof

text: of

text: Theorem

text: Assume

text: that

text: λ

text: is

text: small

text: enough

text: in

text: order

text: to

text: ensure

text: that

text: the

text: conclusion

text: of

text: Lemma

text: holds.

text: By

text: Proposition

text: we

text: have

text: a

text: ﬁrst

text: critical

text: point,

text: which

text: is

text: a

text: local

text: minimum

text: of

text: I

text: On

text: the

text: λ

text: other

text: hand,

text: since

text: the

text: Cerami

text: condition

text: holds,

text: in

text: view

text: of

text: Lemmata

text: and

text: we

text: can

text: apply

text: Theorem

text: and

text: obtain

text: a

text: second

text: critical

text: point

text: of

text: I

text: at

text: the

text: mountain-pass

text: level.

text: This

text: gives

text: two

text: diﬀerent

text: solutions

text: of

text: (Q

text: λ

text: Finally,

text: by

text: Lemma

text: we

text: obtain

text: two

text: solutions

text: of

text: (P

text: λ

text: Proof

text: of

text: Theorems

text: and

text: In

text: this

text: section,

text: we

text: assume

text: the

text: stronger

text: assumption

text: (A

text: In

text: that

text: case,

text: we

text: are

text: able

text: to

text: improve

text: our

text: results

text: on

text: the

text: non-coercive

text: case.

text: Proposition

text: Assume

text: that

text: (A

text: holds

text: with

text: h

text: Then,

text: for

text: every

text: λ

text: there

text: exists

text: v

text: for

text: some

text: τ

text: with

text: v

text: which

text: is

text: a

text: local

text: minimum

text: of

text: I

text: in

text: the

text: and

text: a

text: solution

text: of

text: (Q

text: λ

text: λ

text: with

text: v

text: α

text: (with

text: α

text: deﬁned

text: by

text: λ

text: λ

text: Proof.

text: First

text: of

text: all,

text: observe

text: that,

text: as

text: h

text: we

text: have

text: m

text: and

text: hence,

text: by

text: Theorem

text: (P

text: has

text: a

text: solution

text: u

text: L

text: (Ω).

text: By

text: Lemma

text: p

text: L

text: (Ω),

text: is

text: then

text: a

text: weak

text: solution

text: of

text: µ

text: v

text: v

text: h(x)

text: in

text: Ω,

text: p

text: p

text: v

text: on

text: ∂Ω.

text: As

text: moreover,

text: −µ

text: v

text: L∞(Ω),

text: it

text: follows

text: from

text: that

text: v

text: for

text: some

text: τ

text: and,

text: p

text: by

text: the

text: strong

text: maximum

text: principle

text: (see

text: that

text: v

text: Now,

text: we

text: split

text: the

text: rest

text: of

text: the

text: proof

text: in

text: three

text: steps.

text: Step

text: is

text: a

text: strict

text: upper

text: solution

text: of

text: (Q

text: λ

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 1 f(n) f(n + 1) f(n + 2) f(n + 3) f(n + 4) f(n + 5) − · · q0n 1 24 1 23 − · · − · · q1n 1 212 18 27 1 21 1 20 · · · · q2n 1 29 33 210 13 27 24 25 · · · · · q3n 153 28 1 216 5 212 177 29 3 29 · − · · · · q4n 93 211 1 217 89 213 3 214 3 213 · · · − · q5n 3 217 3 217 89 215 1 218 − · · · · · q6n 93 216 3 218 177 217 5 219 1 222 − · · · · q7n 153 218 24 219 13 220 33 222 − · · − · · − · q8n 1 224 1 220 1 220 18 225 1 229 · − · q9n 1 227 1 227 Table 2: Coeﬃcients of the bimonic q-recurrence for J−pretz(n); for space reasons 2 only the evaluations for q = 2 are given. In order to reveal the underlying symmetry, common powers of q are kept as powers of 2; for example, the entry · 24 25 in the last-but-one column comes from the coeﬃcient of q2nf(n+4) which − is q5(q4 + q3 q + 2). The ﬁrst column corresponds to the inhomogeneous part. ≤ ≤ ≤ ≤ has the following palindromicity properties (0 i 5, 0 j 9). − − − − (cid:96) = q5j i 20(cid:96) − − and (cid:96)− = q5j 25(cid:96)− − . i,j 5 i,9 j 1,j 1,10 j This phenomenon is illustrated in Table 2. It is also visible in Figure 3 but on a diﬀerent example. The occurrence of palindromic operators in the context of knot theory has been studied in more detail in [15]. Indeed, if we use the − ≤ bimonic recurrence to deﬁne the sequence f(n) = (qn 1)J−pretz(n) for n 0 2 then we see that this sequence is palindromic: − − ∈ N f(n) = qn f( n) for all n . We have applied Algorithm 5.4 to all recurrences associated to the twist − ≤ ≤ knots Ktwist for 14 p 15 and to some of the pretzel knots Kpretz. All p p these results can be found in the supplementary electronic material [22]. 6 Conclusion In this paper, we determine a generating set of the q-Weyl closure of a given univariate q-diﬀerence operator, and compute a desingularized operator whose leading coeﬃcient has minimal degree in q. Moreover, we use our algorithms to certify that several instances of the colored Jones polynomial are Laurent poly- nomial sequences. A challenging topic for future research would be to consider the corresponding problems in the multivariate case. Another direction of research we want to consider in the future is the desin- gularization problem for linear Mahler equations [23], which attracted quite some interest in the computer algebra community recently, see for example [8]. Mahler equations arise in the study of automatic sequences, in the complex- ity analysis of divide-and-conquer algorithms, and in some number-theoretic questions. 19 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: f(n)

text: f(n

text: f(n

text: f(n

text: f(n

text: f(n

text: Table

text: Coeﬃcients

text: of

text: the

text: bimonic

text: q-recurrence

text: for

text: J−pretz(n);

text: for

text: space

text: reasons

text: only

text: the

text: evaluations

text: for

text: q

text: are

text: given.

text: In

text: order

text: to

text: reveal

text: the

text: underlying

text: symmetry,

text: common

text: powers

text: of

text: q

text: are

text: kept

text: as

text: powers

text: of

text: for

text: example,

text: the

text: entry

text: in

text: the

text: last-but-one

text: column

text: comes

text: from

text: the

text: coeﬃcient

text: of

text: which

text: is

text: q

text: The

text: ﬁrst

text: column

text: corresponds

text: to

text: the

text: inhomogeneous

text: part.

text: has

text: the

text: following

text: palindromicity

text: properties

text: i

text: j

text: i

text: and

text: i,j

text: j

text: j

text: This

text: phenomenon

text: is

text: illustrated

text: in

text: Table

text: It

text: is

text: also

text: visible

text: in

text: Figure

text: but

text: on

text: a

text: diﬀerent

text: example.

text: The

text: occurrence

text: of

text: palindromic

text: operators

text: in

text: the

text: context

text: of

text: knot

text: theory

text: has

text: been

text: studied

text: in

text: more

text: detail

text: in

text: Indeed,

text: if

text: we

text: use

text: the

text: bimonic

text: recurrence

text: to

text: deﬁne

text: the

text: sequence

text: f(n)

text: (qn

text: for

text: n

text: then

text: we

text: see

text: that

text: this

text: sequence

text: is

text: palindromic:

text: N

text: f(n)

text: qn

text: f(

text: n)

text: for

text: all

text: n

text: We

text: have

text: applied

text: Algorithm

text: to

text: all

text: recurrences

text: associated

text: to

text: the

text: twist

text: knots

text: Ktwist

text: for

text: p

text: and

text: to

text: some

text: of

text: the

text: pretzel

text: knots

text: Kpretz.

text: All

text: p

text: p

text: these

text: results

text: can

text: be

text: found

text: in

text: the

text: supplementary

text: electronic

text: material

text: Conclusion

text: In

text: this

text: paper,

text: we

text: determine

text: a

text: generating

text: set

text: of

text: the

text: q-Weyl

text: closure

text: of

text: a

text: given

text: univariate

text: q-diﬀerence

text: operator,

text: and

text: compute

text: a

text: desingularized

text: operator

text: whose

text: leading

text: coeﬃcient

text: has

text: minimal

text: degree

text: in

text: q.

text: Moreover,

text: we

text: use

text: our

text: algorithms

text: to

text: certify

text: that

text: several

text: instances

text: of

text: the

text: colored

text: Jones

text: polynomial

text: are

text: Laurent

text: poly-

text: nomial

text: sequences.

text: A

text: challenging

text: topic

text: for

text: future

text: research

text: would

text: be

text: to

text: consider

text: the

text: corresponding

text: problems

text: in

text: the

text: multivariate

text: case.

text: Another

text: direction

text: of

text: research

text: we

text: want

text: to

text: consider

text: in

text: the

text: future

text: is

text: the

text: desin-

text: gularization

text: problem

text: for

text: linear

text: Mahler

text: equations

text: which

text: attracted

text: quite

text: some

text: interest

text: in

text: the

text: computer

text: algebra

text: community

text: recently,

text: see

text: for

text: example

text: Mahler

text: equations

text: arise

text: in

text: the

text: study

text: of

text: automatic

text: sequences,

text: in

text: the

text: complex-

text: ity

text: analysis

text: of

text: divide-and-conquer

text: algorithms,

text: and

text: in

text: some

text: number-theoretic

text: questions.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: K (cid:54) (i) If p(x) is an irreducible polynomial in (q)[x] of positive degree with p(0) = ∈ Z 0, so is p(qαx) for each α . K (ii) Let p(x) be an irreducible polynomial in (q)[x] of positive degree with (cid:54) p(0) = 0. Then ∈ Z\{ } gcd(p(qαx), p(x)) = 1 for each α 0 . K (cid:54) (iii) Let f(x), g(x) be two polynomials in (q)[x] with f(0) = 0. Then the set { ∈ N | } α deg (gcd(f(qαx), g(x))) > 0 (4) x is a ﬁnite set. Proof. (i) It follows from [24, Proposition 3]. ∈ Z\{ } (ii) Suppose that there exists α 0 such that 0 (cid:54) gcd(p(qα0x), p(x)) = 1. K | Since p(x) is an irreducible polynomial in (q)[x], we have that p(x) p(qα0x). We may write − · · · p(x) = c xd + c − xd 1 + + c , (5) d d 1 0 ∈ K ≤ ≤ (cid:54) where c (q), 0 i d with d > 0, and c , c = 0. Then i 0 d − − · · · p(qα0x) = (c qdα0)xd + (c − q(d 1)α0)xd 1 + + c . (6) d d 1 0 | Since p(x) p(qα0x), we conclude from (5) and (6) that p(qα0x) = qdα0p(x). Comparing the constant coeﬃcients of both sides in the above equation, it follows that c qdα0 = c . 0 0 (cid:54) Since c0 = 0, weKhave that qdα0 = 1, a contradiction to the fact that q is not a root of unity of . (iii) Suppose that (4) is an inﬁnite set. Then there exists an irreducible factor p(x) of f(x) such that (cid:54) ∈ N gcd(p(qαx), g(x)) = 1 for inﬁnitely many α . Since g(x) only has ﬁnitely many distinct irreducible factors, it follows from (i) that (cid:54) (cid:54) ∈ N gcd(p(qα1x), p(qα2x)) = 1 for some α = α . 1 2 Therefore, we have − (cid:54) gcd(p(qα1 α2x), p(x)) = 1, a contradiction to (ii). 4 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: K

label: list

text: (i)

text: If

text: p(x)

text: is

text: an

text: irreducible

text: polynomial

text: in

text: (q)[x]

text: of

text: positive

text: degree

text: with

text: Z

text: so

text: is

text: p(qαx)

text: for

text: each

text: α

text: K

text: (ii)

text: Let

text: p(x)

text: be

text: an

text: irreducible

text: polynomial

text: in

text: (q)[x]

text: of

text: positive

text: degree

text: with

text: Then

text: Z\{

text: gcd(p(qαx),

text: p(x))

text: for

text: each

text: α

text: K

text: (iii)

text: Let

text: f(x),

text: g(x)

text: be

text: two

text: polynomials

text: in

text: (q)[x]

text: with

text: Then

text: the

text: set

label: equation

text: N

text: α

text: deg

text: (gcd(f(qαx),

text: g(x)))

text: x

text: is

text: a

text: ﬁnite

text: set.

text: Proof.

text: (i)

text: It

text: follows

text: from

text: Proposition

text: Z\{

text: (ii)

text: Suppose

text: that

text: there

text: exists

text: α

text: such

text: that

text: p(x))

text: K

text: Since

text: p(x)

text: is

text: an

text: irreducible

text: polynomial

text: in

text: (q)[x],

text: we

text: have

text: that

text: p(x)

text: We

text: may

text: write

text: p(x)

text: c

text: xd

text: c

text: xd

text: c

text: d

text: d

text: K

text: where

text: c

text: (q),

text: i

text: d

text: with

text: d

text: and

text: c

text: c

text: Then

text: i

text: d

text: (c

text: (c

text: q(d

text: c

text: d

text: d

text: Since

text: p(x)

text: we

text: conclude

text: from

text: and

text: that

text: Comparing

text: the

text: constant

text: coeﬃcients

text: of

text: both

text: sides

text: in

text: the

text: above

text: equation,

text: it

text: follows

text: that

text: c

text: c

text: Since

text: weKhave

text: that

text: a

text: contradiction

text: to

text: the

text: fact

text: that

text: q

text: is

text: not

text: a

text: root

text: of

text: unity

text: of

text: (iii)

text: Suppose

text: that

text: is

text: an

text: inﬁnite

text: set.

text: Then

text: there

text: exists

text: an

text: irreducible

text: factor

text: p(x)

text: of

text: f(x)

text: such

text: that

text: N

text: gcd(p(qαx),

text: g(x))

text: for

text: inﬁnitely

text: many

text: α

text: Since

text: g(x)

text: only

text: has

text: ﬁnitely

text: many

text: distinct

text: irreducible

text: factors,

text: it

text: follows

text: from

text: (i)

text: that

text: N

text: for

text: some

text: α

text: α

text: Therefore,

text: we

text: have

text: p(x))

text: a

text: contradiction

text: to

text: (ii).

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:80) (cid:80) (ii) Let f(x) = d a xi and g(x) = d b xi be two polynomials such that i=0 i ∈ N i=0 i f(qαx) = qeg(x) for some e . Then a b qdα = 0 d . b a (cid:80) 0 d (cid:54) Proof. (i) Assume that f(x) = d a xi with a , a = 0, gcd(a , . . . , a ) = 1 K i=0 i d 0 d 0 in [q]. Then − − · · · f(qαx) = (a qdα)xd + (a − q(d 1)α)xd 1 + + a . (7) d d 1 0 K Since gcd(a , . . . , a ) = 1 in [q], we have that d 0 − ∈ N gcd(a qdα, a − q(d 1)α, . . . , a ) = qe for some e . d d 1 0 Thus, we can write f(qαx) = qeg(x), where g(x) is a primitive polynomial K (cid:54) in [q][x] with the same degree as f(x) and g(0) = 0. (ii) Since f(qαx) = qeg(x), it follows from (7) that a qeb 0 = 0 a qdα qeb d d b = 0 . b d Thus, we conclude that a b qdα = 0 d . b a 0 d ∈ K Given f(x), g(x) (q)[x], we may further assume that f(x), g(x) are two K polynomials in [q][x] by clearing their denominators. The above proposition gives a method to compute the dispersion of two primitive irreducible polyno- K mials in [q][x]. Below is the corresponding algorithm. ∈ K Algorithm 3.4. Given two primitive irreducible polynomials f, g [q][x] of (cid:54) positive degrees with respect to x and f(0) = 0. Compute dis(f, g). (cid:54) (1) Compute(cid:80)d1 = degx(f), d2 =(cid:80)degx(g). If d1 = d2, then return 0. Other- wise, set d = d . 1 (2) Let f = di=0 aixi and g = di=0 bixi. If ab00abdd is not a nonnegative power of qd, then return 0. Otherwise, set α to be the natural number such that qdα = a0bd . b0ad − (3) Compute h = f(qαx) g(x). If h is not the zero polynomial, return 0. adqdα bd Otherwise, return α. The termination of the above algorithm is obvious. The correctness follows from Proposition 3.3. 6 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: (ii)

text: Let

text: f(x)

text: d

text: a

text: xi

text: and

text: g(x)

text: d

text: b

text: xi

text: be

text: two

text: polynomials

text: such

text: that

text: i

text: N

text: i

text: f(qαx)

text: qeg(x)

text: for

text: some

text: e

text: Then

text: a

text: b

text: qdα

text: d

text: b

text: a

text: d

text: Proof.

text: (i)

text: Assume

text: that

text: f(x)

text: d

text: a

text: xi

text: with

text: a

text: a

text: gcd(a

text: a

text: K

text: i

text: d

text: d

text: in

text: [q].

text: Then

text: f(qαx)

text: (a

text: qdα)xd

text: (a

text: q(d

text: a

text: d

text: d

text: K

text: Since

text: gcd(a

text: a

text: in

text: [q],

text: we

text: have

text: that

text: d

text: N

text: gcd(a

text: qdα,

text: a

text: q(d

text: a

text: qe

text: for

text: some

text: e

text: d

text: d

text: Thus,

text: we

text: can

text: write

text: f(qαx)

text: qeg(x),

text: where

text: g(x)

text: is

text: a

text: primitive

text: polynomial

text: K

text: in

text: [q][x]

text: with

text: the

text: same

text: degree

text: as

text: f(x)

text: and

text: (ii)

text: Since

text: f(qαx)

text: qeg(x),

text: it

text: follows

text: from

text: that

text: a

text: qeb

text: a

text: qdα

text: qeb

text: d

text: d

text: b

text: b

text: d

text: Thus,

text: we

text: conclude

text: that

text: a

text: b

text: qdα

text: d

text: b

text: a

text: d

text: K

text: Given

text: f(x),

text: g(x)

text: (q)[x],

text: we

text: may

text: further

text: assume

text: that

text: f(x),

text: g(x)

text: are

text: two

text: K

text: polynomials

text: in

text: [q][x]

text: by

text: clearing

text: their

text: denominators.

text: The

text: above

text: proposition

text: gives

text: a

text: method

text: to

text: compute

text: the

text: dispersion

text: of

text: two

text: primitive

text: irreducible

text: polyno-

text: K

text: mials

text: in

text: [q][x].

text: Below

text: is

text: the

text: corresponding

text: algorithm.

text: K

text: Algorithm

text: Given

text: two

text: primitive

text: irreducible

text: polynomials

text: f,

text: g

text: [q][x]

text: of

text: positive

text: degrees

text: with

text: respect

text: to

text: x

text: and

text: Compute

text: dis(f,

text: g).

text: degx(f),

text: If

text: then

text: return

text: Other-

text: wise,

text: set

text: d

text: d

text: Let

text: f

text: aixi

text: and

text: g

text: bixi.

text: If

text: is

text: not

text: a

text: nonnegative

text: power

text: of

text: qd,

text: then

text: return

text: Otherwise,

text: set

text: α

text: to

text: be

text: the

text: natural

text: number

text: such

text: that

text: qdα

text: Compute

text: h

text: f(qαx)

text: g(x).

text: If

text: h

text: is

text: not

text: the

text: zero

text: polynomial,

text: return

text: adqdα

text: bd

text: Otherwise,

text: return

text: α.

text: The

text: termination

text: of

text: the

text: above

text: algorithm

text: is

text: obvious.

text: The

text: correctness

text: follows

text: from

text: Proposition

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: K Example 3.5. Consider the following two primitive polynomials in [q][x]: − f(x) = qx2 1, − g(x) = q5x2 1. Using the above algorithm, we ﬁnd that dis(f(x), g(x)) = 2. K Using the irreducible factorization over [q][x], we derive the following al- K gorithm to compute the dispersion of two arbitrary polynomials in [q][x]: ∈ K (cid:54) Algorithm 3.6. Given f(x), g(x) [q][x] with f(0) = 0, compute dis(f, g). (1) [Initialize] If deg (f) < 1 or deg (g) < 1 then return 0. Otherwise, set x x dispersion = 0. { } { } (2) [Factorization] Compute the set f (x) and g (x) of distinct primitive K i j irreducible factors over [q] of positive degree in x for f(x) and g(x), respectively. (3) For each pair (f (x), g (x)) of these factors, use Proposition 3.3 to compute i j α = dis(fi(x), gj(x)). If α > dispersion, then set dispersion = α. (4) Return dispersion. The termination of the above algorithm is obvious. The correctness follows from Deﬁnition 3.2 and Proposition 3.3. It is implemented in Mathematica. K Example 3.7. Consider the following two polynomials in [q][x]: − − f(x) = (qx 1)(qx + 1)(qx2 1), − − g(x) = q9x7(q2x 1)(q2x + 1)(q5x2 1). They are already in factored form. Using the above algorithm, we ﬁnd that dis(f(x), g(x)) = 2. 4 Desingularization in the q-Weyl algebra We are now going to present algorithms for the q-Weyl closure (Algorithm 4.10) and for the desingularization of a q-diﬀerence operator (Algorithm 4.13). These algorithms are analogs of those in [28] and use Gr¨obner basis computations. Hence, in practice, they are slower than algorithms based on linear algebra [5, 7] (see also Section ), but their advantage is that also the degree with respect to q can be taken into account—a feature that will be essential for the examples presented in the next section. In this section, we consider the desingularization for the leading coeﬃcient of a given q-diﬀerence operator. The trailing coeﬃcient can be handled in a similar way. We summarize some terminologies given in [5, 7, 28] by specializing the general Ore ring setting to the q-Weyl algebra. ∈ K Deﬁnition 4.1. Let P (q)[x][∂] with positive order, and p be a divisor K of lc (P) in (q)[x]. ∂ 7

text: K

text: Example

text: Consider

text: the

text: following

text: two

text: primitive

text: polynomials

text: in

text: [q][x]:

text: f(x)

text: g(x)

text: Using

text: the

text: above

text: algorithm,

text: we

text: ﬁnd

text: that

text: dis(f(x),

text: g(x))

text: K

text: Using

text: the

text: irreducible

text: factorization

text: over

text: [q][x],

text: we

text: derive

text: the

text: following

text: al-

text: K

text: gorithm

text: to

text: compute

text: the

text: dispersion

text: of

text: two

text: arbitrary

text: polynomials

text: in

text: [q][x]:

text: K

text: Algorithm

text: Given

text: f(x),

text: g(x)

text: [q][x]

text: with

text: compute

text: dis(f,

text: g).

text: [Initialize]

text: If

text: deg

text: (f)

text: or

text: deg

text: (g)

text: then

text: return

text: Otherwise,

text: set

text: x

text: x

text: dispersion

text: [Factorization]

text: Compute

text: the

text: set

text: f

text: (x)

text: and

text: g

text: (x)

text: of

text: distinct

text: primitive

text: K

text: i

text: j

text: irreducible

text: factors

text: over

text: [q]

text: of

text: positive

text: degree

text: in

text: x

text: for

text: f(x)

text: and

text: g(x),

text: respectively.

text: For

text: each

text: pair

text: (f

text: (x),

text: g

text: (x))

text: of

text: these

text: factors,

text: use

text: Proposition

text: to

text: compute

text: i

text: j

text: α

text: dis(fi(x),

text: gj(x)).

text: If

text: α

text: dispersion,

text: then

text: set

text: dispersion

text: α.

text: Return

text: dispersion.

text: The

text: termination

text: of

text: the

text: above

text: algorithm

text: is

text: obvious.

text: The

text: correctness

text: follows

text: from

text: Deﬁnition

text: and

text: Proposition

text: It

text: is

text: implemented

text: in

text: Mathematica.

text: K

text: Example

text: Consider

text: the

text: following

text: two

text: polynomials

text: in

text: [q][x]:

text: f(x)

text: (qx

text: g(x)

text: They

text: are

text: already

text: in

text: factored

text: form.

text: Using

text: the

text: above

text: algorithm,

text: we

text: ﬁnd

text: that

text: dis(f(x),

text: g(x))

text: Desingularization

text: in

text: the

text: q-Weyl

text: algebra

text: We

text: are

text: now

text: going

text: to

text: present

text: algorithms

text: for

text: the

text: q-Weyl

text: closure

text: (Algorithm

text: and

text: for

text: the

text: desingularization

text: of

text: a

text: q-diﬀerence

text: operator

text: (Algorithm

text: These

text: algorithms

text: are

text: analogs

text: of

text: those

text: in

text: and

text: use

text: Gr¨obner

text: basis

text: computations.

text: Hence,

text: in

text: practice,

text: they

text: are

text: slower

text: than

text: algorithms

text: based

text: on

text: linear

text: algebra

text: (see

text: also

text: Section

text: but

text: their

text: advantage

text: is

text: that

text: also

text: the

text: degree

text: with

text: respect

text: to

text: q

text: can

text: be

text: taken

text: into

text: account—a

text: feature

text: that

text: will

text: be

text: essential

text: for

text: the

text: examples

text: presented

text: in

text: the

text: next

text: section.

text: In

text: this

text: section,

text: we

text: consider

text: the

text: desingularization

text: for

text: the

text: leading

text: coeﬃcient

text: of

text: a

text: given

text: q-diﬀerence

text: operator.

text: The

text: trailing

text: coeﬃcient

text: can

text: be

text: handled

text: in

text: a

text: similar

text: way.

text: We

text: summarize

text: some

text: terminologies

text: given

text: in

text: by

text: specializing

text: the

text: general

text: Ore

text: ring

text: setting

text: to

text: the

text: q-Weyl

text: algebra.

text: K

text: Deﬁnition

text: Let

text: P

text: (q)[x][∂]

text: with

text: positive

text: order,

text: and

text: p

text: be

text: a

text: divisor

text: K

text: of

text: lc

text: (P)

text: in

text: (q)[x].