Elimination of unknowns for systems of algebraic differential-difference equations
Elimination of unknowns for systems of algebraic differential-difference equations
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DOI：
10.1090/tran/8219
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发表时间：
2018-12
影响因子：
1.3
通讯作者：
Wei Li;A. Ovchinnikov;G. Pogudin;T. Scanlon
中科院分区：
文献类型：
--
作者：
Wei Li;A. Ovchinnikov;G. Pogudin;T. Scanlon
我们为常微分差分方程建立了有效的消去定理。具体来说，我们找到自然数参数 r r 和 s s 的可计算函数 B ( r , s ) B(r,s) ，因此对于变量 x = x 1 , … , x q \mathbfit {x} = x_1, \ldots , x_q 和 y = y 1 , … , y r \mathbfit {y} = 中的任何代数常微分方程组y_1, \ldots , y_r ，其中每一个都在 y \mathbfit {y} 中具有阶和度，以 s s 为界，在微分差值域上，该系统存在一个仅涉及 x \mathbfit {x} 变量的不平凡的结果，当且仅当这样的结果可以通过应用不超过 B ( r , s ) B(r,s) 次基本差和的迭代来代数构造时系统方程的导数算子。我们将这个有限定理与寻找函数环中的微分差分方程组的解的问题联系起来，表明 C \mathbb {C} 上的微分差分方程组是代数一致的，当且仅当它在亚纯函数的某个胚芽环中具有解。
We establish effective elimination theorems for ordinary differential-difference equations. Specifically, we find a computable function B ( r , s ) B(r,s) of the natural number parameters r r and s s so that for any system of algebraic ordinary differential-difference equations in the variables x = x 1 , … , x q \mathbfit {x} = x_1, \ldots , x_q and y = y 1 , … , y r \mathbfit {y} = y_1, \ldots , y_r , each of which has order and degree in y \mathbfit {y} bounded by s s over a differential-difference field, there is a nontrivial consequence of this system involving just the x \mathbfit {x} variables if and only if such a consequence may be constructed algebraically by applying no more than B ( r , s ) 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 C \mathbb {C} is algebraically consistent if and only if it has solutions in a certain ring of germs of meromorphic functions.