In this paper, we revisit the problem of computing a cylindrical algebraic decomposition from a more geometric perspective, where the construction is related to the study of morphisms between real varieties. It is showed that the geometric fiber cardinality (geometric property) decides the existence of semi-algebraic continuous sections (semi-algebraic property). As a result, all equations can be systematically exploited in the projection phase, leading to a new simple algorithm whose efficiency is demonstrated by experimental results.

In this paper, we establish a new criterion for covering maps between real algebraic varieties. Specifically, we prove that a quasi-finite, flat morphism with locally constant geometric fibers between varieties over a real closed field induces a covering map on the real points in the Euclidean topology. This result provides an effective method for verifying covering properties, as we demonstrate that the required conditions can be checked algorithmically using the tools developed in this work.

This paper studies stratifications of the discriminant hypersurface. We showed that for the general monic univariate polynomial of fixed degree n, its higher branch loci in the coefficient space defined by sub-discriminants can be found by the branch locus defined by the discriminant.

This paper proposes a new family of morphisms between varieties, namely the q-étale morphisms (= flat + finitely and constantly many geometric points in fibers). It is shown in the paper that q-étale morphisms become finite étale after reduction, therefore induce covering maps on the real points for real varieties.

This is a short explanation of the work "a geometric approach to cylindrical algebraic decomposition", firstly presented in ISSAC'24 poster session.

This paper shows that reduction can be widely performed in a class of problems called "Trigonometric Extension".

This paper is about a real root isolation algorithm for rational univariate mixed trigonometric-polynomials.

This paper is about the first-order theory of univariate mixed trigonometric-polynomials. We showed that this theory is surprisingly unconditionally decidable.