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Convex bodies and multiplicities of ideals. (English) Zbl 1315.13013
Proc. Steklov Inst. Math. 286, 268-284 (2014) and Tr. Mat. Inst. Steklova 286, 291-307 (2014).
The connection between multiplicities of monomial ideals in a polynomial ring and convex geometry is known by the work of Bernstein and Kouchnirenko. The aim of this article is to extend this connection to a more general setting.
Let $$(R, {\mathfrak m})$$ be an analytically irreducible local domain of dimension $$n$$, and $${\mathbf k}:=R/{\mathfrak m}$$ the residue field. In this article it is proved that $$R$$ has a “good” valuation, that is, a valuation $$v$$ with values in $${\mathbb Z}^n$$, satisfying some technical properties. On the other hand, let $$({\mathfrak a}_\bullet )$$ be a $${\mathfrak m}-$$ primary sequence of subspaces of $${\mathfrak m}$$, such that $${\mathfrak a}_k {\mathfrak a}_m\subset {\mathfrak a}_{k+m}$$ for any natural integers $$k,m$$, and $${\mathfrak a}_1$$ contains a power of $${\mathfrak m}$$. The authors prove that the limit: $e({\mathfrak a}_\bullet ):=n!\lim_{k\rightarrow +\infty } \frac{\text{dim}_{\mathbf k}(R/{\mathfrak a}_k)}{k^n}$ exists. The number $$e({\mathfrak a}_\bullet )$$ is called the multiplicity of $$({\mathfrak a}_\bullet)$$.
Let $$C\subset {\mathbb R}^n$$ be the closure of the convex hull of the value semigroup $$S$$ of the valuation $$v$$. The authors define a convex region $$\Gamma ({\mathfrak a}_\bullet)\subset C$$ such that $$C\setminus \Gamma ({\mathfrak a}_\bullet)$$ is bounded.
The main result in this paper is that $$e({\mathfrak a}_\bullet )=n! \text{vol}(C\setminus \Gamma ({\mathfrak a}_\bullet))$$. As a consequence they get the Brunn-Minkowski inequality: $e({\mathfrak a}_\bullet )^{1/n}+ e({\mathfrak b}_\bullet )^{1/n}\geq e({\mathfrak a}_\bullet {\mathfrak b}_\bullet )^{1/n}$

##### MSC:
 13A18 Valuations and their generalizations for commutative rings 13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series 13F55 Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes 14M25 Toric varieties, Newton polyhedra, Okounkov bodies 13D10 Deformations and infinitesimal methods in commutative ring theory
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