Raising coefficients of a power series to some power

Question

Suppose you are given a power series $P=\sum_{i=0}^\infty{a_nt^n}$. I am primarily concerned with those power series coming from rational functions of the form $$ \frac{1}{\prod_{i=1}^k{(1-t^{\alpha_i})}}.$$ My motivation comes from looking at Hilbert series of noetherian freely generated subrings of polynomial rings. In general, I know $P$ and its expression as a rational function. I then consider the power series $P'=\sum_{i=0}^\infty{a_n^\kappa t^n}$ for some $\kappa\in\mathbb{Z}_{>0}$. I am interested in determining facts about the rational function that describes this power series (there should be one as it is a Hilbert series of a ring). Of greatest interest to me is the degree of the polynomial appearing in the denominator.

Answer 1

It's a general fact about any rational power series

$$\sum_{n=0}^\infty a_n t^n = {p(t) \over \prod_{i=1}^r (1-\gamma_i t)^{d_i}}$$

(where $p(t)$ is some polynomial of degree less than $\sum_{i=1}^r d_i$ and the $\gamma_i$ are distinct) that

$$a_n = \sum_{i=1}^r p_i(n)\gamma_i^n$$

for some polynomials $p_i$ of degree less than $d_i$. Conversely, any $a_n$ that can be expressed this way are the coefficients of a rational power series. See for example Theorem 4.1.1 in Richard Stanley's book Enumerative Combinatorics. From this it is immediate that $a_n^\kappa$ are the coefficients of a rational power series, as you say, and moreover by chasing through the proof of the theorem you can construct the denominator that you are interested in. I think that this shows that the degree will go up by at most a factor of $\kappa$, but to compute the degree exactly, you may have to keep careful track of which roots of unity occur with what multiplicities.