Solving parametric polynomial systems using Generic Rational Univariate Representation

2026/02/06 by Florent Corniquel · 1 voice
Computer Science · #cs.SC

paper · pdf · doi:10.48550/arxiv.2602.06817

Abstract

In this paper, we present a generic parametrization of generically zero-dimensional parametric polynomial systems. More specifically, we study the specialization properties of the Rational Univariate Representation and derive bounds on the degrees and heights of its elements. In addition to that, we propose two algorithms to effectively compute this parametrization.

Citations

• Reading Rational Univariate Representations on lexicographic Groebner bases
• Heights of varieties in multiprojective spaces and arithmetic Nullstellensatze
• On the asymptotic and practical complexity of solving bivariate systems over the reals
• Complexity of Resolution of Parametric Systems of Polynomial Equations and Inequations
• On the computational complexity and geometry of the first-order theory of the reals. Part I: Introduction. Preliminaries. The geometry of semi-algebraic sets. The decision problem for the existential theory of the reals
• Real quantifier elimination is doubly exponential
• Some Formulae in Elimination
• Algorithms in Real Algebraic Geometry

Discussions

• Florent Corniquel (SU, UPCit\'e, IMJ-PRG, OURAGAN): Solving parametric polynomial systems using Generic Rational Univariate Representation https://arxiv.org/abs/2602.06817 https://arxiv.org/pdf/2602.0 [bsky, 0 points, 0 comments]

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• A note on Solving Parametric Polynomial Systems
• Generic Regular Decompositions for Parametric Polynomial Systems
• Fast Rational Univariate Representation via Gaussian Elimination
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• Reading Rational Univariate Representations on lexicographic Groebner bases