## View abstract

### Session S07 - Differential operators in algebraic geometry and commutative algebra

Friday, July 16, 14:30 ~ 15:00 UTC-3

## Valuative aspects of differential algebraic geometry

### 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('cloak063c1cb0c6cc92f2160daecb964a2689').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy063c1cb0c6cc92f2160daecb964a2689 = 'cr&#105;sth&#105;&#97;n.g&#97;r&#97;y' + '&#64;'; addy063c1cb0c6cc92f2160daecb964a2689 = addy063c1cb0c6cc92f2160daecb964a2689 + 'c&#105;m&#97;t' + '&#46;' + 'mx'; var addy_text063c1cb0c6cc92f2160daecb964a2689 = 'cr&#105;sth&#105;&#97;n.g&#97;r&#97;y' + '&#64;' + 'c&#105;m&#97;t' + '&#46;' + 'mx';document.getElementById('cloak063c1cb0c6cc92f2160daecb964a2689').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy063c1cb0c6cc92f2160daecb964a2689 + '\'>'+addy_text063c1cb0c6cc92f2160daecb964a2689+'<\/a>';

Broadly speaking, the key ingredient for tropicalization is a generalized non-Archimedean absolute value; that is, is a function defined on a commutative ring with unit that satisfies properties analogous to those of a non-Archimedean absolute value (i.e., Krull valuation), but for which we allow the target to be a commutative idempotent semiring.

In this talk we will introduce non-Krull absolute values for rings of multivariate formal power series (and their fields of fractions) which are defined in terms of Newton polyhedra and polytopes associated to their supports in $\mathbb{N}^m$. We show how to use them to study systems of classical partial algebraic differential equations from a valuative (tropical) point of view.