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arXiv · 1712.01412

Effective difference elimination and Nullstellensatz

Alexey Ovchinnikov·Gleb Pogudin·Thomas Scanlon
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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 $\mathbf{x} = (x_1, \ldots, x_m)$ and $\mathbf{u} = (u_1, \ldots, u_r)$, 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.

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Alexey Ovchinnikov, Gleb Pogudin, Thomas Scanlon. 2017-12-04. Effective difference elimination and Nullstellensatz. https://doi.org/10.4171/jems/968

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