# Effective difference elimination and Nullstellensatz

### Alexey Ovchinnikov

CUNY Queens College, USA### Gleb Pogudin

New York University, USA### Thomas Scanlon

University of California at Berkeley, USA

## Abstract

We prove effective Nullstellensatz and elimination theorems for difference equations in sequence rings. More precisely, we compute an explicit function of geometric quantities associated to a system of difference equations (and these geometric quantities may themselves be bounded by a function of the number of variables, the order of the equations, and the degrees of the equations) so that for any system of difference equations in variables $x=(x_{1},…,x_{m})$ and $u=u_{1},…,u_{r})$, if these equations have any nontrivial consequences in the $x$ variables, then such a consequence may be seen algebraically considering transforms up to the order of our bound. Specializing to the case of $m=0$, we obtain an effective method to test whether a given system of difference equations is consistent.

## Cite this article

Alexey Ovchinnikov, Gleb Pogudin, Thomas Scanlon, Effective difference elimination and Nullstellensatz. J. Eur. Math. Soc. 22 (2020), no. 8, pp. 2419–2452

DOI 10.4171/JEMS/968