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# Journal of the European Mathematical Society

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**Volume 22, Issue 8, 2020, pp. 2419–2452**

**DOI: 10.4171/JEMS/968**

Published online: 2020-04-17

Effective difference elimination and Nullstellensatz

Alexey Ovchinnikov^{[1]}, Gleb Pogudin

^{[2]}and Thomas Scanlon

^{[3]}(1) CUNY Queens College, USA

(2) New York University, USA

(3) University of California at Berkeley, USA

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 $\mathbf x = (x_1, \dots, x_m)$ and $\mathbf u = u_1, \dots, u_r)$, if these equations have any nontrivial consequences in the $\mathbf x$ 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.

*Keywords: *Difference equations, effective Nullstellensatz, elimination of unknowns

Ovchinnikov Alexey, Pogudin Gleb, Scanlon Thomas: Effective difference elimination and Nullstellensatz. *J. Eur. Math. Soc.* 22 (2020), 2419-2452. doi: 10.4171/JEMS/968