JournalsjemsVol. 22, No. 8pp. 2419–2452

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
Effective difference elimination and Nullstellensatz cover

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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=(x1,,xm)\mathbf x = (x_1, \dots, x_m) and u=u1,,ur)\mathbf u = u_1, \dots, u_r), if these equations have any nontrivial consequences in the x\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=0m = 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