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Speaker: Li Wei, associate researcher, Academy of Mathematics and Systems Science, Chinese Academy of Sciences
Date: October 27, 2022
Location: Tencent Meeting
Sponsor: School of Mathematics, Shandong University
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.
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