On Parametric Linear System Solving

Mathematics > Rings and Algebras [Submitted on 29 Aug 2025 (v1), last revised 15 Jun 2026 (this version, v2)] Title:On Parametric Linear System Solving View PDF HTML (experimental)Abstract:Parametric linear systems are linear systems of equations in which some symbolic parameters, that is, symbols that are not considered to be candidates for elimination or solution in the course of analyzing the problem, appear in the coefficients of the system. In this paper we assume that the symbolic parameters appear polynomially in the coefficients and that the only variables to be solved for are those of the linear system. The consistency of the system and expression of the solutions may vary depending on the values of the parameters. It is well-known that it is possible to specify a covering set of regimes, each of which is a Zariski-constructible condition on the parameters together with a solution description valid under that condition. We provide a method of solution that requires time polynomial in the matrix dimension and the degrees of the polynomials when there are up to three parameters. We also discuss examples suggesting how the method may be useful beyond the formal three-parameter setting. In previous methods the number of regimes needed is exponential in the system dimension and polynomial degree of the parameters. Our approach exploits the Hermite and Smith normal forms that may be computed when the system coefficient domain is mapped to the univariate polynomial domain over suitably constructed fields. Our method identifies {intrinsic singularities} and {ramification points} where the algebraic and geometric structure of the matrix changes. Parametric eigenvalue problems are addressed as well. Submission history From: Robert Corless [view email][v1] Fri, 29 Aug 2025 13:43:16 UTC (25 KB) [v2] Mon, 15 Jun 2026 18:39:55 UTC (53 KB) Current browse context: math.RA References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer (What is the Explorer?) Connected Papers (What is Connected Papers?) Litmaps (What is Litmaps?) scite Smart Citations (What are Smart Citations?) Code, Data and Media Associated with this Article alphaXiv (What is alphaXiv?) CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub (What is DagsHub?) Gotit.pub (What is GotitPub?) Hugging Face (What is Huggingface?) ScienceCast (What is ScienceCast?) Demos Recommenders and Search Tools Influence Flower (What are Influence Flowers?) CORE Recommender (What is CORE?) arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.