T. Oaku, N. Takayama, H. Tsai
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.
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A function that is analytic on a domain of $\mathbb{C}^n$ is holonomic if it is the solution to a holonomic system of linear homogeneous differential equations with polynomial coefficients. We define and study the Bernstein-Sato polynomial of a holonomic function on a smooth algebraic variety. We analyze the structure of certain sheaves of holonomic functions, such as the algebraic functions along a hypersurface, determining their direct sum decompositions into indecomposables, that further respect decompositions of Bernstein-Sato polynomials. When the space is endowed with the action of a lin…
In algebraic geometry, one studies the solutions to polynomial equations, or, equivalently, to linear partial differential equations with constant coefficients. These lecture notes address the more general case when the coefficients are polynomials. The letter D stands for the Weyl algebra, and a D-module is a left module over D. We focus on left ideals, or D-ideals. We represent holonomic functions in several variables by the linear differential equations they satisfy. This encoding by a D-ideal is useful for many problems, e.g., in geometry, physics and statistics. We explain how to work wit…
Local holomorphic solutions z=z(a) to a univariate sparse polynomial equation p(z) =0, in terms of its vector of complex coefficients a, are classically known to satisfy holonomic systems of linear partial differential equations with polynomial coefficients. In this paper we investigate one of such systems of differential equations which was introduced by Mellin. We compute the holonomic rank of the Mellin system as well as the dimension of the space of its algebraic solutions. Moreover, we construct explicit bases of solutions in terms of the roots of p and their logarithms. We show that the …
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.
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.
A non-zero constant Jacobian polynomial maps $F=(P,Q)$ of $\mathbb{C}^2$ is invertible if $P$ and $Q$ are rational polynomials.
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。
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。
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。
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.
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