{"title":"Simplifying matrix differential equations with general coefficients","authors":"","doi":"10.1007/s11856-023-2599-0","DOIUrl":null,"url":null,"abstract":"<h3>Abstract</h3> <p>We show that the <em>n × n</em> matrix differential equation <em>δ</em>(<em>Y</em>) = <em>AY</em> with <em>n</em><sup>2</sup> general coefficients cannot be simplified to an equation in less than <em>n</em> parameters by using gauge transformations whose coefficients are rational functions in the matrix entries of <em>A</em> and their derivatives. Our proof uses differential Galois theory and a differential analogue of essential dimension. We also bound the minimum number of parameters needed to describe some generic Picard–Vessiot extensions.</p>","PeriodicalId":14661,"journal":{"name":"Israel Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2023-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Israel Journal of Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s11856-023-2599-0","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We show that the n × n matrix differential equation δ(Y) = AY with n2 general coefficients cannot be simplified to an equation in less than n parameters by using gauge transformations whose coefficients are rational functions in the matrix entries of A and their derivatives. Our proof uses differential Galois theory and a differential analogue of essential dimension. We also bound the minimum number of parameters needed to describe some generic Picard–Vessiot extensions.
期刊介绍：
The Israel Journal of Mathematics is an international journal publishing high-quality original research papers in a wide spectrum of pure and applied mathematics. The prestigious interdisciplinary editorial board reflects the diversity of subjects covered in this journal, including set theory, model theory, algebra, group theory, number theory, analysis, functional analysis, ergodic theory, algebraic topology, geometry, combinatorics, theoretical computer science, mathematical physics, and applied mathematics.
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