一、题目：

On the Luroth Problem for Partial Differential Fields

二、主讲人：

李伟

三、摘要：

In 1875, J. Luroth proved the well-known Luroth's theorem, which states that every subfield of the rational function field in one variable is a simple extension. Geometrically, the Luroth's theorem means a unirational curve is always rational. Around 1940s, Ritt and Kolchin proved the ordinary differential version of Luroth's theorem, and thus unirational differential curves rational. But as Kolchin pointed out, the Luroth's theorem ceases to hold for partial differential fields. A natural question arises: under what conditions are unirational partial differential curves rational?

In this talk, we present our partial differential Luroth's theorem in both theoretical and algorithmic aspects. We first give a necessary and sufficient condition for a subfield of a partial differential rational function field to be a simple extension. This result generalizes Ritt and Kolchin’s classical differential Luroth's theorem. We then give an algorithm to decide whether a given finitely generated differential subfield admits a Luroth generator, and in the affirmative case, to compute a Luroth generator. As an application, we solve the problem of deciding whether a unirational partial differential curve is rational.

四、主讲人简介：

李伟，中科院数学与系统科学研究院副研究员。2007年于山东大学获理学学士学位，2012年在中科院数学与系统科学研究院获博士学位。研究方向为构造性微分代数几何，主要成果包括：与合作者建立了微分周形式与微分周簇理论，发展了稀疏微分结式的理论及单指数消元算法，证明了微分-差分方程的有效Hilbert零点定理。论文主要发表在Found. Comput. Math., Trans. Amer. Math. Soc., J. Lond. Math. Soc., J. Symb. Comput.等期刊与ISSAC会议文集。主持国家自然科学基金委优秀青年科学基金等项目；曾获国际计算机协会(ACM) SIGSAM ISSAC 2011唯一杰出论文奖、中科院优秀博士学位论文、中科院数学院“陈景润未来之星”、吴文俊计算机数学青年学者奖等奖项。

五、邀请人：

黄巧龙 数学学院副研究员

六、时间：

10月27日（周四）10:00-11:00

七、地点：

腾讯会议

八、联系人：

黄巧龙，联系方式：huangqiaolong@sdu.edu.cn

九、主办单位：

山东大学数学学院