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Deciding first-order formulas involving univariate mixed trigonometric-polynomials

Published in International Symposium on Symbolic and Algebraic Computation, 2023

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

Recommended citation: Chen, R., & Xia, B. (2023, July). Deciding first-order formulas involving univariate mixed trigonometric-polynomials. In Proceedings of the 2023 International Symposium on Symbolic and Algebraic Computation (pp. 145-154).

Isolating all the real roots of a mixed trigonometric-polynomial

Published in Journal of Symbolic Computation, 2024

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

Recommended citation: Chen, R., Li, H., Xia, B., Zhao, T., & Zheng, T. (2024). Isolating all the real roots of a mixed trigonometric-polynomial. Journal of Symbolic Computation, 121, 102250.

Reduction of Transcendental Decision Problems over the Reals

Published in International Symposium on Symbolic and Algebraic Computation, 2024

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

Recommended citation: Chen, R., & Xia, B. (2024, July). Reduction of Transcendental Decision Problems over the Reals. In Proceedings of the 2024 International Symposium on Symbolic and Algebraic Computation (pp. 56-64).

What Kind of Morphisms Induces Covering Maps over a Real Closed Field?

Published in arXiv preprint, 2025

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.

Recommended citation: Chen, R. (2025). What Kind of Morphisms Induces Covering Maps over a Real Closed Field?. arXiv preprint arXiv:2502.05834.

A Geometric Approach to Cylindrical Algebraic Decomposition

Published in Mathematics of Computation, 2025

This paper is about the geometry behind the cylindrical algebraic decomposition (CAD, a classical construction in real algebraic geometry). We show that this construction is related to studying finite free (flat) morphisms of real varieties and present a new algorithm for CAD.

Recommended citation: Rizeng Chen, A Geometric Approach to Cylindrical Algebraic Decomposition, Math. Comp., electronically published on June 3, 2025, DOI: https://doi.org/10.1090/mcom/4099 (to appear in print).

Stratifying Discriminant Hypersurface

Published in arXiv preprint, 2025

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.

Recommended citation: Chen, R., Hong, H. & Yang, J. (2025). Stratifying Discriminant Hypersurface. arXiv preprint arXiv:2509.25820.