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.

中文翻译：

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有效的差异消除和 Nullstellensatz

我们证明了序列环中差分方程的有效 Nullstellensatz 和消元定理。更准确地说，我们计算与差分方程组相关的几何量的显式函数（并且这些几何量本身可能受变量数量、方程的阶数和方程的阶数的函数的限制），因此对于变量 $\mathbf{x} = (x_1, \ldots, x_m)$ 和 $\mathbf{u} = (u_1, \ldots, u_r)$ 中的任何差分方程组，如果这些方程有任何非平凡的后果在 $\mathbf{x}$ 变量中，那么可以通过代数方式看到这样的结果，考虑到我们界限的阶数的变换。专门针对 $m = 0$ 的情况，我们获得了一种有效的方法来测试给定的差分方程组是否一致。