报告题目：Additive Decompositions in Symbolic Integration

主讲人：李子明 研究员 (中科院数学与系统科学研究院)

报告时间：2023年6月17日（周六）上午8:30-10:30

报告地点：海创(大连)科技交流中心(海创厅)

报告摘要：Symbolic integration aims to develop algorithms for evaluating integrals in finite terms. One of its classical topics is determining whether an integrand has an elementary integral, and computing such an integral if one exists. Fundamental results on this topic are collected and reviewed in [13]. Algorithms for integrating transcendental functions are presented in [3]. Additive decompositions compute “remainders” and determine the so-called “in-field integrability”. More precisely, for a given function f in a differential field F, an additive decomposition computes g, r ∈ F such that f = g′ + r with the following two properties:

(i) (minimality) r is minimal in some technical sense,

(ii) (in-field integrability) f has an integral in F if and only if r = 0.

The Hermite-Ostrogradsky reduction developed in the 19th century is an additive decomposition for rational functions. However, additive decompositions for more sophisticated functions had not yet been developed until the Hermite reduction for hyper exponential functions was introduced in 2013.

We are going to review classical Risch’s algorithm for integrating transcendental functions, and present recent progress in additive decompositions in this talk. Additionally, new results will be discussed on how to combine classical algorithms with additive decompositions so as to enhance and accelerate the integrators implemented in computer algebra systems such as Maple and Mathematica.

This talk is dedicated to a special memorial of Professor Marko Petkovšek. His paper coauthored with Sergei Abramov played a key role in bringing additive decompositions back to symbolic integration.

个人简历：李子明，中科院数学与系统科学研究院研究员，1996年在奥地利开普勒大学符号计算研究所获得博士学位。他把代数子结式理论推广到了非交换的Ore多项式，设计了计算Ore多项式最大右公因子的模算法和计算最小左公倍式的高效算法，被商用符号计算系统Maple采用；设计了确定有限维微分-差分模所有子模的算法；给出了微分-差分混合情形下相容函数的结构定理和混合超几何项Zeilberger算法终止性的充要条件；给出了超指数函数和超几何项的完整的加法分解，是第四代计算邻差算子算法的主要设计者之一。曾获2006年国际计算机协会符号与代数计算委员会颁发的ISSAC2006杰出论文奖和ISSAC2014杰出poster奖。指导的博士生曾获ISSAC杰出学生论文奖，多名学生获得中科院和海外学术机构的双学位。