2
$\begingroup$

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 $i,j$.

$\endgroup$

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.