Let $a(n)$ be linear recurrence with constant coefficient of order $t$.

Assume $a(n)=\sum_{i=0}^t c_i r_i^n$ where $r_i$ are the roots of the companion polynomial and $c_i$ are algebraic numbers.

The zeros of $a(n)$ are finite union of arithmetic progressions, possibly of one element.

Assume $a(n)=0$ and $d$ is divisor of $n$.

Point on the projective variety $V_d : 0=\sum_{i=0}^t c_i x_i^{d}$ is $P_d=(r_1^{\frac{n}{d}},r_2^{\frac{n}{d}}\ldots)$

For all $d$, is it true that:

  1. The union of all $P_d$ are finite number of projective points or
  2. Some proper subset sum of the monomials of $V_d$ vanishes at $P_d$

(1) is related to recurrences with $r_i / r_j$ not a root of unity for some $i,j$.


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