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Elimination of unknowns for systems of algebraic differential-difference equations
Transactions of the American Mathematical Society ( IF 1.3 ) Pub Date : 2020-10-14 , DOI: 10.1090/tran/8219
Wei Li , Alexey Ovchinnikov , Gleb Pogudin , Thomas Scanlon

We establish effective elimination theorems for differential-difference equations. Specifically, we find a computable function $B(r,s)$ of the natural number parameters $r$ and $s$ so that for any system of algebraic differential-difference equations in the variables $\mathbf{x} = x_1, \ldots, x_q$ and $\mathbf{y} = y_1, \ldots, y_r$ each of which has order and degree in $\mathbf{y}$ bounded by $s$ over a differential-difference field, there is a non-trivial consequence of this system involving just the $\mathbf{x}$ variables if and only if such a consequence may be constructed algebraically by applying no more than $B(r,s)$ iterations of the basic difference and derivation operators to the equations in the system. We relate this finiteness theorem to the problem of finding solutions to such systems of differential-difference equations in rings of functions showing that a system of differential-difference equations over $\mathbb{C}$ is algebraically consistent if and only if it has solutions in a certain ring of germs of meromorphic functions.

中文翻译:

消除代数微分方程组的未知数

我们为微分差分方程建立了有效的消元定理。具体来说,我们找到了自然数参数 $r$ 和 $s$ 的可计算函数 $B(r,s)$,因此对于变量 $\mathbf{x} = x_1 中的任何代数微分方程组, \ldots, x_q$ 和 $\mathbf{y} = y_1, \ldots, y_r$ 每一个在 $\mathbf{y}$ 中的阶次和度数以 $s$ 为界,在一个微分差分域上,有这个系统的非平凡结果只涉及 $\mathbf{x}$ 变量当且仅当这样的结果可以通过应用不超过 $B(r,s)$ 的基本差分和推导运算符的迭代代数构造到系统中的方程。
更新日期:2020-10-14
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