Ish. It's a "non-Abelian" Picard-Fuchs equation, coming from isomonodromic deformations of flat SL_2-connections on P^1\{x_1, ..., x_4}. Closely related to Painleve equations...
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Odgovor korisniku/ci @littmath
Cool. Which paper is that?
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Odgovor korisniku/ci @tejotaefe
This one: https://www.cambridge.org/core/services/aop-cambridge-core/content/view/6D01E9B2FA50EC50775D09E27C995D39/S0017089501000106a.pdf/discrete_and_continuous_painleve_vi_hierarchy_and_the_garnier_systems.pdf … It's actually closely related to
@monsoon0's work, I believe!1 reply 1 proslijeđeni tweet 6 korisnika označava da im se sviđa -
Trying to learn about Painleve equations because of something that came up in my work re:the p-curvature conjecture...
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Odgovor korisnicima @littmath @tejotaefe
What is the conjecture? Please say more ...
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Odgovor korisnicima @monsoon0 @tejotaefe
It's involved -- one wants to test whether an algebraic differential equation admits a full set of algebraic solutions by reducing it mod p for almost all p. Brian Lawrence and I gave a (non-trivial) reformulation in terms of isomonodromy here: https://arxiv.org/abs/1907.03941
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Odgovor korisnicima @littmath @tejotaefe
Fascinating. I need to understand what “a full set” means in this setting.
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I can answer this. Suppose you have a linear ODE of order n with coefficients in the field of rational functions with algebraic coefficients. Locally, there are n linearly independent *holomorphic* solutions. When can you take all of them to be *algebraic* functions?
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In general the solutions of an algebraic ODE are not algebraic. But sometimes they are. Grothendieck conjectured an arithmetic criterion to test this.
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This is the original p-curvature conjecture. Then Katz generalized it to vector bundles with connection (and proved the conjecture for equations of "geometric origin"). There's also a non-linear version, for foliations, but evidence is meager and some ppl don't believe it
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Wow. I just wrote an article for Notices of @amermathsoc. It mentions related questions, leading from construction of algebraic functions as solutions of algebraic differential (or discrete) equations and transcendental functions
https://arxiv.org/abs/1912.08959 Hopefully out in June
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I’ve just been reading it :)!
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I love serendipity ...
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