The EMS Publishing House is now EMS Press and has its new home at

Please find all EMS Press journals and articles on the new platform.

Journal of the European Mathematical Society

Full-Text PDF (259 KB) | Metadata | Table of Contents | JEMS summary
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