报 告 人： 侯庆虎 教授
南开大学 博士生导师
报告题目：An Algorithm for Deciding the Summability of Bivariate Rational Functions
报告时间：
报告地点：静远楼1506学术报告厅
主办单位：数学与统计学院、科技处
报告摘要：Let $\Delta_x f(x,y)=f(x+1,y)-f(x,y)$ and $\Delta_y f(x,y)=f(x,y+1)-f(x,y)$ be the difference operators with respect to $x$ and $y$. A rational function $f(x,y)$ is called summable if there exist rational functions $g(x,y)$ and $h(x,y)$ such that $f(x,y)=\Delta_x g(x,y) + \Delta_y h(x,y)$. Recently, Chen and Singer presented a method for deciding whether a rational function is summable. To implement their method in the sense of algorithms, we need to solve two problems. The first is to determine the shift equivalence of two bivariate polynomials.
We solve this problem by presenting an algorithm for computing the dispersion sets of any two bivariate polynomials. The second is to solve a univariate difference equation in an algebraically closed field. By considering the irreducible factorization of the denominator of $f(x,y)$ in a general field, we present a new criterion which requires only finding a rational solution of a bivariate difference equation. This goal can be achieved by deriving a universal denominator of the rational solutions and a degree bound on the numerator. Combining these two algorithms, we can decide the summability of a bivariate rational function
侯庆虎教授简介：
博士，南开大学教授，博士生导师，教育部新世纪优秀人才，国家自然科学基金委优秀青年基金获得者。主要研究方向为代数组合学，包括组合恒等式的机器证明、组合计数、对称函数理论、q级数理论等领域，在组合数学领域最高级别杂志《J. Combin. Theory Ser. A》、符号计算领域最高级别杂志《J. Symbolic Comput.》发表多篇论文。特别是在组合恒等式的机器证明这一领域取得了系列进展。此外曾参与了金融数据分析、全文检索、图像识别等应用课题的研究。作为副主编参与了《中国城市竞争力报告》系列蓝皮书的数据分析与处理，获得“孙冶方”经济学奖。