arXiv subjects

Rizeng Chen

Publications and source records attributed to Rizeng Chen.

5 recordsLinked to original sources

arXiv2026

An Effective Criterion for Covering Maps Between Real Varieties

We prove that a quasi-finite flat morphism with locally constant geometric-fiber cardinality between varieties over a real closed field induces a covering map on the rational points. The algebraic core of the criterion is a characteristic-zero result showing that every such morphism becomes finite \'etale after reduction. Since reduction does not change rational points, the induced map is a covering in the Euclidean topology. We extend the covering conclusion to Boolean combinations of closed subschemes and their complements, and construct a finite stratification separating finite covering families from families with positive-dimensional fibers. Finally, we compute the canonical non-finite, non-finite-flat, and non-finite-\'etale loci, using Gr\"{o}bner bases. This yields effective tests for the hypotheses of the covering criterion. We conclude with applications illustrating these constructions.

math.AG↗

arXiv2025

Stratifying Discriminant Hypersurface

This paper investigates the stratification of the discriminant hypersurface associated with a univariate polynomial via the number of its distinct complex roots. We introduce two novel approaches different from the one based on subdiscriminants. The first approach stratifies the discriminant hypersurface by recursively removing all the lowest-order points, while the second one stratifies the discriminant hypersurface by recursively removing all the smooth points. Both approaches rely solely on the discriminant itself instead of using high-order subdiscriminants. These results offer new insights into the intrinsic geometry of the discriminant and its connection to root multiplicity.

math.AG↗

arXiv2025

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

In this article, we show that a flat morphism of $k$-varieties ($\mathop{\mathrm{char}} k=0$) with locally constant geometric fibers becomes finite \'etale after reduction. When $k$ is a real closed field, we prove that such a morphism induces a covering map on the rational points. We further give a triviality result different from Hardt's and a new interpretation of the construction of cylindrical algebraic decomposition as applications.

math.AG↗

arXiv2023

A Geometric Approach to Cylindrical Algebraic Decomposition

Cylindrical algebraic decomposition is a classical construction in real algebraic geometry. Although there are many algorithms to compute a cylindrical algebraic decomposition, their practical performance is still very limited. In this paper, we revisit this problem from a more geometric perspective, where the construction of cylindrical algebraic decomposition 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.

math.AG↗

arXiv2023

Isolating Bounded and Unbounded Real Roots of a Mixed Trigonometric-Polynomial

Mixed trigonometric-polynomials (MTPs) are functions of the form $f(x,\sin{x}, \cos{x})$ with $f\in\mathbb{Q}[x_1,x_2,x_3]$. In this paper, an algorithm ``isolating" all the real roots of an MTP is provided and implemented. It automatically divides the real roots into two parts: one consists of finitely many ``bounded" roots in an interval $[μ_-,μ_+]$ while the other consists of probably countably many ``periodic" roots in $\mathbb{R}\backslash[μ_-,μ_+]$. For bounded roots, the algorithm returns isolating intervals and corresponding multiplicities while for periodic roots, it returns finitely many mutually disjoint small intervals $I_i\subset[-π,π]$, integers $c_i>0$ and multisets of root multiplicity $\{m_{j,i}\}_{j=1}^{c_i}$ such that any periodic root $t>μ_+$ is in the set $(\sqcup_i\cup_{k\in\mathbb{N}}(I_i+2kπ))$ and any interval $I_i+2kπ\subset(μ_+,\infty)$ contains exactly $c_i$ periodic roots with multiplicities $m_{1,i},...,m_{c_i,i}$, respectively. The effectiveness and efficiency of the algorithm are shown by experiments. %In particular, our results indicate that the ``distributions" of the roots of an MTP in the ``periods" $(-π,π]+2kπ$ sufficiently far from $0$ share a same pattern. Besides, the method used to isolate the roots in $[μ_-,μ_+]$ is applicable to any other bounded interval as well. The algorithm takes advantages of the weak Fourier sequence technique and deals with the intervals period-by-period without scaling the coordinate so to keep the length of the sequence short. The new approaches can easily be modified to decide whether there is any root, or whether there are infinitely many roots in unbounded intervals of the form $(-\infty,a)$ or $(a,\infty)$ with $a\in\mathbb{Q}$.

cs.SC↗