Effective difference elimination and Nullstellensatz
Effective difference elimination and Nullstellensatz
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DOI：
10.4171/jems/968
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发表时间：
2017-12
期刊：
影响因子：
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通讯作者：
A. Ovchinnikov;G. Pogudin;T. Scanlon
中科院分区：
文献类型：
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作者：
A. Ovchinnikov;G. Pogudin;T. Scanlon
证明了序列环上差分方程的有效零星定理和消元定理。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（这些几何量本身可能由变量的数量，方程的阶数和方程的阶数的函数限制），因此对于任何变量$\mathbf{x}的差分方程组=（x_1，\ldots，x_m）$和$\mathbf{u} =（u_1，\ldots，u_r）$，如果这些方程在$\mathbf{x}$变量中有任何非平凡的结果，那么这样的结果可以在代数上看到，考虑到我们的界阶的变换。针对$m = 0$的情况，我们得到了检验给定差分方程组是否一致的有效方法。
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.