It is known that linear recurrences with constant coefficients can be computed via powers in $\mathbb{Z}[x]/f(x)$.

We believe that this generalizes to quotients of multivariate polynomial rings.

Let $f_1(x_1,..,x_m),f_2(x_1,..,x_m)...f_k(x_1,...,x_m)$. be polynomials with integer coefficients.

Define the quotient $K=\mathbb{Q}[x_1,...,x_m]/(f_1,...f_k)$.

Let the term order be lexicographic (for an answer your can chose another order).

Working over $K$, let $t(n)=(\sum_{i=1}^m x_i)^n$.

$t(n)$ is group satisfying $t(n+m)=t(n) t(m)$.

In general, if $k=m$ then $t(n)$ is of bounded degree and it is efficiently computable.

Define $a(n)$ to be the coefficient of $x_1$ in $t(n)$.

Q1 When $a(n)$ satisfy linear recurrence with constant rational coefficients?

Very limited experimental data suggests that the probability is about 1/2.