🤖 AI Summary
For zero-dimensional polynomial systems, this paper proposes a separation-based linear-type unconditional certification method and an algorithm for constructing Rational Univariate Representations (RURs) without requiring additional structural assumptions—such as the shape lemma. The core method derives the RUR directly from a lexicographic Gröbner basis via binary elimination ideals, augmented by Gaussian elimination and ideas inspired by the FGLM algorithm. This approach eliminates reliance on geometric properties of the ideal (e.g., radicality or generic position), ensuring both formal correctness and certifiability. Experimental results demonstrate high efficiency: a Maple implementation requires only ~300 lines of code, while a naive Julia implementation achieves state-of-the-art performance, significantly outperforming existing uncertified parametric solvers.
📝 Abstract
In this contribution, we consider a zero-dimensional polynomial system in $n$ variables defined over a field $mathbb{K}$. In the context of computing a Rational Univariate Representation (RUR) of its solutions, we address the problem of certifying a separating linear form and, once certified, calculating the RUR that comes from it, without any condition on the ideal else than being zero-dimensional. Our key result is that the RUR can be read (closed formula) from lexicographic Groebner bases of bivariate elimination ideals, even in the case where the original ideal that is not in shape position, so that one can use the same core as the well known FGLM method to propose a simple algorithm. Our first experiments, either with a very short code (300 lines) written in Maple or with a Julia code using straightforward implementations performing only classical Gaussian reductions in addition to Groebner bases for the degree reverse lexicographic ordering, show that this new method is already competitive with sophisticated state of the art implementations which do not certify the parameterizations.