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](https://doc.sagemath.org/html/en/reference/polynomial_rings/sage/rings/polynomial/term_order.html) 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](https://mathoverflow.net/questions/462043/rings-with-many-pairs-of-zero-divisors).

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.