🤖 AI Summary
This work addresses the lack of efficient parametric solution methods for zero-dimensional parametric polynomial systems that exhibit favorable specialization properties. It presents the first systematic study of the specialization behavior of Rational Univariate Representations (RURs) in parametric settings, establishing explicit upper bounds on the degree and height of their constituent elements. By leveraging techniques from algebraic geometry and symbolic computation, the authors develop a general RUR-based parametrization framework and introduce two efficient algorithms. The proposed approach guarantees stable specialization, provides rigorous algebraic complexity bounds, and yields a fully computable and implementable parametric solution method.
📝 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.