Rational polynomials of simple type

Univariate Rational Sums of Squares

Given rational univariate polynomials f and g such that gcd(f, g) and f / gcd(f, g) are relatively prime, we show that g is non-negative on all the real roots of f if and only if g is a sum of squares of rational polynomials modulo f. We complete our study by exhibiting an algorithm that produces a certificate that a polynomial g is non-negative on the real roots of a non-zero polynomial f , when the above assumption is satisfied.

实特殊正交群SO₄(ℝ)上有理曲线的Kempe分解

Kempe factorizations for rational curves on $\operatorname{SO}_4(\mathbb{R})$

该研究针对实特殊正交群SO₄(ℝ)上的有理曲线，受运动多项式分解等启发，通过Cayley分解等方法构造性地完成Kempe分解，所得算法显式并附实例说明。

Plane Jacobian Conjecture for rational polynomials

A non-zero constant Jacobian polynomial maps $F=(P,Q)$ of $\mathbb{C}^2$ is invertible if $P$ and $Q$ are rational polynomials.

Sums of squares of polynomials with rational coefficients

We construct families of explicit polynomials f with rational coefficients that are sums of squares of polynomials over the real numbers, but not over the rational numbers. Whether or not such examples exist was an open question originally raised by Sturmfels. We also study representations of f as sums of squares of rational functions with rational coefficients. In the case of ternary quartics, we prove that our counterexamples to Sturmfels' question are the only ones.

Multivariable polynomial injections on rational numbers

For each number field k, the Bombieri-Lang conjecture for k-rational points on surfaces of general type implies the existence of a polynomial f(x,y) in k[x,y] inducing an injection k x k --> k.

Plane Jacobian problem for rational polynomials

This paper has been withdrawn by the author due to a crucial error in the last lines in the proof of Lemma 3.3.

具有有理系数的正多项式的平方和分解

Sum of squares decomposition of positive polynomials with rational coefficients

本文构造了一个有理系数正多项式，它在实数域上可分解为平方和但在有理数域上不可，并验证其定义非奇异射影超曲面，解答了Scheiderer提出的两个开放问题。

Polynomial and rational solutions of holonomic systems

The aim of this paper is to give two new algorithms, which are elimination free, to find polynomial and rational solutions for a given holonomic system associated to a set of linear differential operators in the Weyl algebra D = k<x_1, ..., x_n, dx_1, ..., dx_n> where k is a subfield of the complex numbers.

单位球内稳定的多项式与有界的有理函数

Stable polynomials and bounded rational functions in the unit ball

研究单位球内无零点的多项式及其对有理函数有界性的贡献，通过局部描述和高维特征分析，探讨有理函数在球上的有界性问题。

具有可界定比特大小的加权平方和多项式的有理对偶证书

Rational Dual Certificates for Weighted Sums-of-Squares Polynomials with Boundable Bit Size

本文研究加权平方和多项式有理对偶证书的比特大小复杂度，给出一般性界并分析一元多项式等特殊情形，同时提出一个有理算术算法来计算具有可界定比特大小的证书。

Dual certificates and efficient rational sum-of-squares decompositions for polynomial optimization over compact sets

We study the problem of computing weighted sum-of-squares (WSOS) certificates for positive polynomials over a compact semialgebraic set. Building on the theory of interior-point methods for convex optimization, we introduce the concept of dual cone certificates, which allows us to interpret vectors from the dual of the sum-of-squares cone as rigorous nonnegativity certificates of a WSOS polynomial. Whereas conventional WSOS certificates are alternative representations of the polynomials they certify, dual certificates are distinct from the certified polynomials; moreover, each dual certificate…

Enumeration of almost polynomial rational functions with given critical values

Enumerating ramified coverings of the sphere with fixed ramification types is a well-known problem first considered by A. Hurwitz. Up to now, explicit solutions have been obtained only for some families of ramified coverings, for instant, those realized by polynomials in one complex variable. In this paper we obtain an explicit answer for a large new family of coverings, namely, the coverings realized by simple almost polynomials, defined below. Unlike most other results in the field, our formula is obtained by elementary methods.

一个计算有理函数消元子式和Bernstein-Sato多项式的算法

An algorithm for annihilator and Bernstein-Sato polynomial of a rational function

本文提出一个算法用于计算有理函数的Bernstein-Sato多项式，通过非交换PBW代数中的Gröbner基实现，支持现有猜想并展示非平凡例子。

Flips of moduli spaces and transition formulas for Donaldson polynomial invariants of rational surfaces

We study the change of moduli spaces of Gieseker-semistable torsion free rank-$2$ sheaves on algebraic surfaces as we vary the polarizations. When the surfaces are rational with an effective anti-canonical divisor, the moduli spaces are linked by a series of flips (blowups and blowdowns). Using these results, we compute the transition formulas for Donaldson polynomial invariants of rational surfaces. Part of the work is also obtained independently by Matsuki-Wentworth and Ellingsrud-G{\" o}ttsche.

Weil polynomials of abelian varieties over finite fields with many rational points

We consider the finite set of isogeny classes of $g$-dimensional abelian varieties defined over the finite field $\mathbb{F}_q$ with endomorphism algebra being a field. We prove that the class within this set whose varieties have maximal number of rational points is unique, for any prime even power $q$ big enough and verifying mild conditions. We describe its Weil polynomial and we prove that the class is ordinary and cyclic outside the primes dividing an integer that only depends on $g$. In dimension $3$, we prove that the class is ordinary and cyclic and give explicitly its Weil polynomial, …

Counting rational points of quiver moduli

It is shown that rational points over finite fields of moduli spaces of stable quiver representations are counted by polynomials with integer coefficients. These polynomials are constructed recursively using an identity in the Hall algebra of a quiver.

Galois点与具有小值集的有理函数

Galois points and rational functions with small value sets

本研究建立了Galois点与有限域上小值集有理函数的联系，证明含双Galois点的平面曲线定义多项式的相关性质，打通了两个研究方向的关联。

多变量稳定多项式与有界有理函数的局部理论

Local theory of stable polynomials and bounded rational functions of several variables

本文基于齐次分解、Puiseux展开和传递函数实现提供稳定多项式的局部理论，借此证明多圆盘有界有理函数的边界非切向极限，并刻画双上半平面分子理想与双圆盘可积性。

一种将半正定多项式表示为有理系数平方和的新算法

A Novel Algorithm for Representing Positive Semi-Definite Polynomials as Sums of Squares with Rational Coefficients

针对现有半正定多项式SOS分解多为浮点系数的问题，提出一种逐步约简技术，可构造保持有理系数的梯状平方和分解，适用于精确运算场景，实验验证其优于现有数值方法。