极小对合基
Minimal Involutive Bases
AI总结：
本文提出构造极小对合多项式基的算法，定义两种新对合除法并证明其性质，确保算法正确终止，且极小对合基在给定序下唯一，可作典范基。
摘要：
本文提出了一种构造极小对合多项式基的算法，这类基是特殊形式的格罗布纳基。最一般的对合算法基于对合单项式除法的概念，该概念将变量划分为乘法变量和非乘法变量。这种划分由此通过执行非乘法延拓和乘法约化，为构造对合基提供了一个自洽的计算过程。每个特定的对合除法都会生成一种特定形式的对合计算过程。除了Thomas、Janet和Pommaret在偏微分方程分析中使用的三种对合除法外，我们还定义了两种新的对合除法。这两种除法以及Thomas除法都不依赖于变量的顺序。我们证明了新除法的诺特性、连续性和构造性，这保证了对于任何有限输入多项式集合和任何可接受的单项式序，对合算法的正确性和终止性。我们表明，给定一个可接受的单项式序，一个首一的极小对合基是唯一确定的，因此可以像约化格罗布纳基一样被视为典范基。
英文摘要：
In this paper we present an algorithm for construction of minimal involutive polynomial bases which are Groebner bases of the special form. The most general involutive algorithms are based on the concept of involutive monomial division which leads to partition of variables into multiplicative and non-multiplicative. This partition gives thereby the self-consistent computational procedure for constructing an involutive basis by performing non-multiplicative prolongations and multiplicative reductions. Every specific involutive division generates a particular form of involutive computational procedure. In addition to three involutive divisions used by Thomas, Janet and Pommaret for analysis of partial differential equations we define two new ones. These two divisions, as well as Thomas division, do not depend on the order of variables. We prove noetherity, continuity and constructivity of the new divisions that provides correctness and termination of involutive algorithms for any finite set of input polynomials and any admissible monomial ordering. We show that, given an admissible monomial ordering, a monic minimal involutive basis is uniquely defined and thereby can be considered as canonical much like the reduced Groebner basis.
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