Solving parametric polynomial systems using Generic Rational Univariate Representation
Florent Corniquel
TL;DR
This work extends the Rational Univariate Representation (RUR) to parametric polynomial systems by introducing Generic Rational Univariate Representations (GRUR). It proves that, for almost all parameter values and a generically separating linear form, the GRUR computed over the parameter field specializes to a classical RUR of the specialized system, and it derives degree and height bounds using arithmetic Nullstellensätze. It presents two Las-Vegas algorithms to compute GRUR: a linear-algebra based method leveraging trace matrices and a second evaluation/interpolation method (GRUR-EI) with explicit complexity analysis and interpolation guarantees. The paper also details real root classification via Sturm-Habicht sequences and includes an appendix on Stickelberger’s theorem and supporting sub-algorithms. Collectively, the results enable certified, parametric solving and analysis across parameter spaces with provable complexity and correctness properties.
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.