零环上的无限小切16次希尔伯特问题

IF 1.3 3区数学 Q2 MATHEMATICS, APPLIED

Bulletin des Sciences Mathematiques Pub Date : 2025-04-04 DOI:10.1016/j.bulsci.2025.103634

J.L. Bravo , P. Mardešić , D. Novikov , J. Pontigo-Herrera

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引用次数: 0

摘要

本文给出两个单变量多项式f和g，以及f的一个0循环C，考虑变形f+ϵg。我们定义了两个函数：位移函数Δ（t, λ）和它的一阶近似：阿贝尔积分M1(t)。零环的无穷小和切向16阶希尔伯特问题分别是计算Δ（t, λ）、λ small或M1(t)的孤立正则零的问题。我们证明了这两个问题是不等价的，并找到了零环上的无限小切向16阶Hilbert问题的最优界，它是f和g度的函数。这两个问题是平面上矢量场的经典无限小问题和切向16阶希尔伯特问题的零维模拟。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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Infinitesimal and tangential 16-th Hilbert problem on zero-cycles

In this paper, given two polynomials f and g of one variable and a 0-cycle C of f, we consider the deformation

f + ϵ g

. We define two functions: the displacement function

Δ (t, ϵ)

and its first order approximation: the abelian integral

M_{1} (t)

The infinitesimal and tangential 16-th Hilbert problem for zero-cycles are problems of counting isolated regular zeros of

Δ (t, ϵ)

, for ϵ small, or of

M_{1} (t)

, respectively.

We show that the two problems are not equivalent and find optimal bounds, in function of the degrees of f and g, for the infinitesimal and tangential 16-th Hilbert problem on zero-cycles. These two problems are the zero-dimensional analog of the classical infinitesimal and tangential 16-th Hilbert problems for vector fields in the plane.

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