Effective difference elimination and Nullstellensatz

TL;DR

This paper develops effective bounds and algorithms for difference equations, enabling explicit computation of consequences and consistency testing based on geometric and algebraic properties of the systems.

Contribution

It introduces effective Nullstellensatz and elimination theorems for difference equations with explicit bounds based on geometric quantities.

Findings

Provides explicit bounds for difference equations consequences.

Enables algebraic testing of system consistency.

Connects geometric properties to algebraic solvability.

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.