Effective difference elimination and Nullstellensatz

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 $\mathbf{x}=(x_1,\dots ,x_m)$ 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.

Cited by in corpus (5)

• Invariant sets and nilpotency of endomorphisms of algebraic sofic shifts
• On the dimension of systems of algebraic difference equations
• Elimination of unknowns for systems of algebraic differential-difference equations
• New bounds and efficient algorithm for sparse difference resultant
• Linear difference operators with sequence coefficients having infinite-dimentional solution spaces