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### Session S24 - Symbolic Computation: Theory, Algorithms and Applications

Tuesday, July 13, 13:00 ~ 13:25 UTC-3

## Initial degenerations for systems of algebraic differential equations

### Cristhian Garay López

#### Center for Research in Mathematics (CIMAT), México   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloakb8c5db4e05150e4056989f0379b79efc').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addyb8c5db4e05150e4056989f0379b79efc = 'cr&#105;sth&#105;&#97;n.g&#97;r&#97;y' + '&#64;'; addyb8c5db4e05150e4056989f0379b79efc = addyb8c5db4e05150e4056989f0379b79efc + 'c&#105;m&#97;t' + '&#46;' + 'mx'; var addy_textb8c5db4e05150e4056989f0379b79efc = 'cr&#105;sth&#105;&#97;n.g&#97;r&#97;y' + '&#64;' + 'c&#105;m&#97;t' + '&#46;' + 'mx';document.getElementById('cloakb8c5db4e05150e4056989f0379b79efc').innerHTML += '<a ' + path + '\'' + prefix + ':' + addyb8c5db4e05150e4056989f0379b79efc + '\'>'+addy_textb8c5db4e05150e4056989f0379b79efc+'<\/a>';

By a degeneration, we mean a process that transforms a geometric object $X/F$ defined over a field $F$ into a simpler object that retains many of the relevant properties of $X$. Formally, any degeneration is realized by an integral model for $X$; that is, a flat scheme $X’/R$ defined over some integral domain $R$ whose generic fiber is the original object $X$.

In this talk, we endow the field $F=K(\!(t_1,\ldots,t_m)\!)$ of quotients of  multivariate formal power series with a  generalized non-Archimedean absolute value $|\cdot|$, and we establish the existence of integral models over the  ring of integers $R=\{|x|\leq 1\}$ for solutions $X$ of systems of algebraic partial differential equations with coefficients on $F$. We also concretely describe the specialization map of a model $X’/R$ to the maximal ideals of $R$, which are encoded in terms of monomial orderings.