TITLE: What are polynomial like expressions that have complex numbers for the exponents?
QUESTION [9 upvotes]: Got curious about polynomials and Galois theory the other day and realized I have no idea how current mathematics treats polynomials (or rather polynomial like expressions) that have arbitrary algebras for the exponents. A quick search yields polynomial extensions like Laurent polynomials but I couldn't find anything that uses any other groups for the exponent, like Gaussian integers, complex numbers, cyclotomic rings, hypercomplex numbers and other algebras.
What are these structures called and how is their behavior similar and different with ordinary polynomials?

REPLY [3 votes]: If $R$ is any ring and $G$ is any group, then the group ring $R[G]$ is ring whose elements are finite formal sums of terms labelled by elements $g\in G$ and with coefficients in$~R$. If the group $G$ is written additively, then one can write such terms as $rX^g$ where $X$ is a formal indeterminate, since multiplication of terms is given by $rX^g*sX^h=rsX^{g+h}$; this resembles polynomials with coefficients in $R$ and exponents in$~G$. If however $G$ is already written multiplicatively, it is better to write terms simply as $r.g$. The construction is also valid for a monoid in the role of$~G$ and is then called a monoid ring; the case of the additive monoid$~\Bbb N$ gives the usual polynomials over$~R$.
The main difference with usual polynomials, or with Laurent polynomials, is that there is no such thing as evaluating a group ring element $\sum_i r_iX^{g_i}$ at some value $X=a\in R$. The reason is that there is not in general any way to define exponentiation $a^g$ in a way consistent with the multiplication of formal terms, namely such that $a^ga^h=a^{g+h}$ for all possible values $g,h\in G$, when $G$ goes beyond the set of natural numbers or (in case $a$ is invertible) the set of integers. Problems already appear in such simple cases as $G=\Bbb Q$ (the additive group), even with "nice" coefficient fields $R=\Bbb R$ or $R=\Bbb C$: there is no consistent way to define rational powers for negative reals or of general complex numbers so that the usual laws of exponentiation are valid in all cases. In fact one can for $R=\Bbb C$ salvage just the rule $z^az^b=z^{a+b}$ (even with $a,b\in\Bbb C$) that does not require any relation between evaluations at different values, by choosing a branch of the complex logarithm and putting $z^a=\exp(a\ln z)$ for this choice; extending this linearly does define a ring morphism $\Bbb C[\Bbb Q]\to\Bbb C$ of "evaluation of polynomials with rational exponents at a fixed number$~z$". Even so, the corresponding "polynomial function" of$~z$ cannot be made to be continuous on$~\Bbb C$, and there will also be failure of other algebraic identities that one may expect, such as $z^{-a}=(z^{-1})^a$ for $z\neq0$ (taking $z=-1$ and $a=\frac12$ shows the impossibility to have this) or $(xy)^a=x^ay^a$. You can find many questions on this site related to this fundamental impossibility to define exponentiation for not-positive-real base and non-integer exponent in a way consistent with the usual rules manipulating powers (and roots, viewed as fractional powers).
Of course going beyond $G=\Bbb Q$ things only get more problematic. Since evaluation is one of the most important things we do with polynomials, this makes working in group or monoid rings rather different than working with polynomials, and for this reason one does not call monoid ring elements polynomials although they are formally defined in much the same way. Note however that $R[\Bbb N^k]\cong R[X_1,\ldots,X_k]$, so in this case the definition amounts to that of (multivariate) polynomials, for which evaluation (in $k$-tuples of values) is well defined.