This article is cited in 25 papers

Convex bodies and multiplicities of ideals

Kiumars Kaveh^a, Askold Khovanskii<sup>bcd

a</sup> Department of Mathematics, School of Arts and Sciences, University of Pittsburgh, 301 Thackeray Hall, Pittsburgh, PA 15260, USA
^b Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, Ontario, M5S 2E4 Canada
^c Independent University of Moscow, Bol'shoi Vlas'evskii per. 11, Moscow, 119002 Russia
^d Institute for Systems Analysis, Russian Academy of Sciences, pr. 60-letiya Oktyabrya 9, Moscow, 117312 Russia

Abstract: We associate convex regions in $\mathbb R^n$ to $\mathfrak m$-primary graded sequences of subspaces, in particular $\mathfrak m$-primary graded sequences of ideals, in a large class of local algebras (including analytically irreducible local domains). These convex regions encode information about Samuel multiplicities. This is in the spirit of the theory of Gröbner bases and Newton polyhedra on the one hand, and the theory of Newton–Okounkov bodies for linear systems on the other hand. We use this to give a new proof as well as a generalization of a Brunn–Minkowski inequality for multiplicities due to Teissier and Rees–Sharp.

UDC: 514.172.45

Received in April 2013

Language: English

DOI: 10.1134/S0371968514030169

 English version: Proceedings of the Steklov Institute of Mathematics, 2014, 286, 268–284

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