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Amoebas, Monge-Ampère measures and triangulations of the Newton polytope
Stockholm University, Faculty of Science, Department of Mathematics. Matematik.
2004 (English)In: Duke Mathematical Journal, Vol. 121, no 3, 481-507 p.Article in journal (Refereed) Published
Abstract [en]

The amoeba of a holomorphic function $f$ is, by definition, the image in $\mathbf{R}^n$ of the zero locus of $f$ under the simple mapping that takes each coordinate to the logarithm of its modulus. The terminology was introduced in the 1990s by the famous (biologist and) mathematician Israel Gelfand and his coauthors Kapranov and Zelevinsky (GKZ). In this paper we study a natural convex potential function $N_f$ with the property that its Monge-Ampére mass is concentrated to the amoeba of $f$ We obtain results of two kinds; by approximating $N_f$ with a piecewise linear function, we get striking combinatorial information regarding the amoeba and the Newton polytope of $f$; by computing the Monge-Ampére measure, we find sharp bounds for the area of amoebas in $\mathbf{R}^n$. We also consider systems of functions $f_{1},\dots,f_{n}$ and prove a local version of the classical Bernstein theorem on the number of roots of systems of algebraic equations.

Place, publisher, year, edition, pages
2004. Vol. 121, no 3, 481-507 p.
National Category
Mathematics
Identifiers
URN: urn:nbn:se:su:diva-20272DOI: doi:10.1215/S0012-7094-04-12134-7OAI: oai:DiVA.org:su-20272DiVA: diva2:186798
Available from: 2007-11-23 Created: 2007-11-23 Last updated: 2011-01-12Bibliographically approved

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