Daniel Daigle
Let k be an algebraically closed field. A polynomial F in k[X,Y] is said to be "generally rational" if, for almost all c in k, the curve " F= c '' is rational. It is well known that, if char(k)=0, F is generally rational iff there exists G in k(X,Y) such that k(F,G)=k(X,Y). We give analogous results valid in arbitrary characteristic.
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We classify two-variable polynomials which are rational of simple type. These are precisely the two-variable polynomials with trivial homological monodromy.
We determine the Newton trees of the rational polynomials of simple type, thus filling a gap in the proof of the classification of these polynomials given by Neumann and Norbury.
We propose a new lifting and recombination scheme for rational bivariate polynomial factorization that takes advantage of the Newton polytope geometry. We obtain a deterministic algorithm that can be seen as a sparse version of an algorithm of Lecerf, with now a polynomial complexity in the volume of the Newton polytope. We adopt a geometrical point of view, the main tool being derived from some algebraic osculation criterions in toric varieties.
It has been proved several times in the literature that a polynomial map from $C^2$ to $C$ with irreducible rational fibers cannot be a component of a counterexample to the Jacobian Conjecture. This note points out that this result is empty: it is implicit in 1980 work of Miyanish and Sugie that such a polynomial is equivalent to $f(x,y)=x$ by a polynomial automorphism of $C^2$.
Effective bounds for polynomial systems defined over the rationals
本文综述了有理数域上多项式系统的有效界，重点探讨了有限次元代数簇的估计方法与结果。
This paper is withdrawn. The current main theorem can be proved by using a simple field theory. The main theorem is to placed by another theorem, shortly.
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.
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。
A non-zero constant Jacobian polynomial maps $F=(P,Q)$ of $\mathbb{C}^2$ is invertible if $P$ and $Q$ are rational polynomials.
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.
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.
This paper has been withdrawn by the author due to a crucial error in the last lines in the proof of Lemma 3.3.
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。
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.
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。
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。
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…
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.
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。
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.
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, …
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 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。
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。
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。
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