Generalized characteristic sets and new multivariate difference dimension polynomials

IF 0.6 4区工程技术 Q4 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS

Applicable Algebra in Engineering Communication and Computing Pub Date : 2023-10-28 DOI:10.1007/s00200-023-00628-0

Alexander Levin

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引用次数: 0

Abstract

We introduce a new type of characteristic sets of difference polynomials using a generalization of the concept of effective order to the case of partial difference polynomials and a partition of the basic set of translations \(\sigma\). Using properties of these characteristic sets, we prove the existence and outline a method of computation of a multivariate dimension polynomial of a finitely generated difference field extension that describes the transcendence degrees of intermediate fields obtained by adjoining transforms of the generators whose orders with respect to the components of the partition of \(\sigma\) are bounded by two sequences of natural numbers. We show that such dimension polynomials carry essentially more invariants (that is, characteristics of the extension that do not depend on the set of its difference generators) than previously known difference dimension polynomials. In particular, a dimension polynomial of the new type associated with a system of algebraic difference equations gives more information about the system than the classical univariate difference dimension polynomial.

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广义特征集和新的多元差维多项式

我们利用有效阶概念对部分差分多项式和基本平移集分割的概括，引入了差分多项式的新型特征集。利用这些特征集的性质，我们证明了有限生成差分域扩展的多元维度多项式的存在并概述了计算方法，它描述了通过对生成器的邻接变换得到的中间域的超越度，这些生成器相对于 \(\sigma\) 分区的分量的阶是由两个自然数序列限定的。我们证明，与之前已知的差分维多项式相比，这种维多项式本质上携带了更多的不变式（即不依赖于其差分生成子集的扩展特征）。特别是，与代数差分方程系统相关联的新类型维度多项式比经典的单变量差分维度多项式提供了更多关于该系统的信息。

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来源期刊

Applicable Algebra in Engineering Communication and Computing 工程技术-计算机：跨学科应用

CiteScore

2.90

自引率

14.30%

发文量

审稿时长

>12 weeks

期刊介绍： Algebra is a common language for many scientific domains. In developing this language mathematicians prove theorems and design methods which demonstrate the applicability of algebra. Using this language scientists in many fields find algebra indispensable to create methods, techniques and tools to solve their specific problems. Applicable Algebra in Engineering, Communication and Computing will publish mathematically rigorous, original research papers reporting on algebraic methods and techniques relevant to all domains concerned with computers, intelligent systems and communications. Its scope includes, but is not limited to, vision, robotics, system design, fault tolerance and dependability of systems, VLSI technology, signal processing, signal theory, coding, error control techniques, cryptography, protocol specification, networks, software engineering, arithmetics, algorithms, complexity, computer algebra, programming languages, logic and functional programming, algebraic specification, term rewriting systems, theorem proving, graphics, modeling, knowledge engineering, expert systems, and artificial intelligence methodology. Purely theoretical papers will not primarily be sought, but papers dealing with problems in such domains as commutative or non-commutative algebra, group theory, field theory, or real algebraic geometry, which are of interest for applications in the above mentioned fields are relevant for this journal. On the practical side, technology and know-how transfer papers from engineering which either stimulate or illustrate research in applicable algebra are within the scope of the journal.