TITLE: Are there three non-commutative polynomials in three variables with finite dimensional quotient?
QUESTION [11 upvotes]: $\newcommand\la{\langle}\newcommand\ra{\rangle}$Let $K$ be a field and $K\la x,y,z\ra$ the non-commutative polynomial ring in 3 variables.

Question 1: Are there three (fewer is probably not possible?!) polynomials $f,g,h \in K\la x,y,z\ra$, which are sums of monomials of degree at least two, such that the algebra $K\la x,y,z\ra/(f,g,h)$ is finite dimensional?

Is there a systematic way to construct such polynomials?
For two variables we can take for example $f=xy+yx$ and $g=x^3+y^3$, but I do not know (or forgot) such examples for more than 2 variables.
(If possible we should also have (this is equivalent to $(f,g,h)$ being an admissible ideal) that there exists an $n \geq 2$ such that $J^n \subseteq (f,g,h)$ where $J=\la x,y,z\ra$ is the ideal generated by $x,y,z$.)

Question 2: Let $f_i$ for $i=1,\dotsc,n-1$ be polynomials spanning an admissible ideal in the non-commutative polynomial ring in $n-1$ variables $x_i$. Can we find $g$ such that $\la f_i,g\ra$ is an admissible ideal in $K\la x_i,y\ra$?

REPLY [7 votes]: Q1. Yes, such 3 monomials exist. Moreover, the generic homogeneous quadratic polynomials $f,g,h$ are such that the dimension of the quotient algebra $K⟨x,y,z⟩/(f,g,h)$ is 28. This follows, in particular, from Th.1.3 of [Natalia Iyudu and Stanislav Shkarin. The Golod-Shafarevich inequality for Hilbert series of
quadratic algebras and the Anick conjecture].
Q2. In contrast, such an ideal does not exist if $n\ge 5$. If follows from the Vinberg version of the Golod--Shafarevich theorem that if an ideal $I$ is generated by $r$ elements (which are linear combinations of at least quadratic monomials) in a free associative algebra $F$ with $g$ generators, then the quotient algebra is infinite-dimensional provided that the polynomial $1-gz+rz^2$ has a positive root (cf, for example, Prop. 2.7 in [M.Ershov, Golod-Shafarevich groups: a survey]). In your case we have $r=g=n-1$, so that the quotient algebra is infinite if $n\ge 5$.