Seminar: Dialogue Across Three Centuries---Newton Polygon, Legendre Transform, and a p-adic Jensen Formula

Seminar| Institute of Mathematical Sciences

Time:Friday, May 23th, 2025,13:30-15:00

Abstract:The Jensen formula gives the exact relationship between the number of zeros of an entire function on the complex plane at a given radius and the integral of the average of its logarithmic modulus; replacing the complex field with a valued field leads to an analogous zero-counting problem: how to count the zeros of a function of prescribed valuation. Around 350 years before Jensen’s work, Newton introduced the tool now known as the “Newton polygon” in his study of the Puiseux series, whose key property is that the polygon of a product is the convolution of the polygons of its factors. I will review how the Legendre transforms links these two problems, then discuss the geometric analog of the complex plane in the p-adic setting and explain the p-adic Jensen formula. The talk will conclude by showing how this framework can be used to investigate the basic algebraic structure of certain important rings, such as obtaining an estimate of the Krull dimensions.(Undergraduate students are welcome.)

报告题目：跨越三个世纪的对话---Newton 多边形、Legendre 变换与 p 进 Jensen 公式

摘要：Jensen 公式揭示了复平面上整函数在给定半径的圆周上零点数与其对数模值平均之积分之间的精确关系；当将复数域替换为赋值域时，就可提出类似的零点计数问题：如何计算函数的具有给定赋值的零点个数。在 Jensen 公式提出前约 350 年，Newton 在研究分式指数幂级数时引入了“Newton 多边形”这一工具，其基本性质是：乘积函数对应的多边形等于各自多边形的卷积。我将回顾如何借助 Legendre 变换将上述两类问题联系起来，进而讨论复平面在 p 进赋值域情形下的对应几何图形，并给出什么是 p 进 Jensen 公式。报告最后将展示如何利用这一框架研究一些重要环的基本代数结构，例如对 Krull 维数的估计。  （欢迎本科生参加。）