子代数与Khovanskii基等价

IF 1.1 4区数学 Q4 COMPUTER SCIENCE, THEORY & METHODS

Journal of Symbolic Computation Pub Date : 2025-08-06 DOI:10.1016/j.jsc.2025.102480

Colin Alstad , Michael Burr , Oliver Clarke , Timothy Duff

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引用次数: 0

摘要

我们研究了两个先前研究过的代数集合中Gröbner基的类似物之间的部分对应：即多项式环商的子代数基和值代数和定义域的Khovanskii基。我们的主要动机是将商环的子代数基的具体和计算方面应用于Khovanskii基的抽象理论。我们的观点是，最有趣的Khovanskii基的例子也可以实现为子代数基，反之亦然。作为这一通信的一部分，我们将多项式环商的子代数基理论推广到无限生成的多项式代数，并研究了使该理论有效的条件。我们还从商环的子代数基数据中提供了牛顿-奥库科夫体的计算，这说明了如何将Khovanskii基解释为子代数基使它们适用于现有的计算机代数工具。

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Subalgebra and Khovanskii bases equivalence

We study a partial correspondence between two previously-studied analogues of Gröbner bases in the setting of algebras: namely subalgebra bases for quotients of polynomial rings and Khovanskii bases for valued algebras and domains. Our main motivation is to apply the concrete and computational aspects of subalgebra bases for quotient rings to the abstract theory of Khovanskii bases. Our perspective is that most interesting examples of Khovanskii bases can also be realized as subalgebra bases and vice-versa. As part of this correspondence, we extend the theory of subalgebra bases for quotients of polynomial rings to infinitely generated polynomial algebras and study conditions which make this theory effective. We also provide a computation of Newton-Okounkov bodies from the data of subalgebra bases for quotient rings, which illustrates how interpreting Khovanskii bases as subalgebra bases makes them amenable to existing computer algebra tools.

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来源期刊

Journal of Symbolic Computation 工程技术-计算机：理论方法

CiteScore

2.10

自引率

14.30%

发文量

审稿时长

142 days

期刊介绍： An international journal, the Journal of Symbolic Computation, founded by Bruno Buchberger in 1985, is directed to mathematicians and computer scientists who have a particular interest in symbolic computation. The journal provides a forum for research in the algorithmic treatment of all types of symbolic objects: objects in formal languages (terms, formulas, programs); algebraic objects (elements in basic number domains, polynomials, residue classes, etc.); and geometrical objects. It is the explicit goal of the journal to promote the integration of symbolic computation by establishing one common avenue of communication for researchers working in the different subareas. It is also important that the algorithmic achievements of these areas should be made available to the human problem-solver in integrated software systems for symbolic computation. To help this integration, the journal publishes invited tutorial surveys as well as Applications Letters and System Descriptions.