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Amoebas, Monge-Ampère measures, and triangulations of the Newton polytope.
Stockholm University, Faculty of Science, Department of Mathematics.
Stockholm University, Faculty of Science, Department of Mathematics.
2004 (English)In: Duke mathematical journal, ISSN 0012-7094, 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.
URN: urn:nbn:se:su:diva-23102OAI: diva2:190165
Part of urn:nbn:se:su:diva-15Available from: 2003-10-02 Created: 2003-10-02 Last updated: 2009-10-07Bibliographically approved
In thesis
1. Topics in geometry, analysis and inverse problems
Open this publication in new window or tab >>Topics in geometry, analysis and inverse problems
2003 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The thesis consists of three independent parts.

Part I: Polynomial amoebas

We study the amoeba of a polynomial, as de ned by Gelfand, Kapranov and Zelevinsky. A central role in the treatment is played by a certain convex function which is linear in each complement component of the amoeba, which we call the Ronkin function. This function is used in two di erent ways. First, we use it to construct a polyhedral complex, which we call a spine, approximating the amoeba. Second, the Monge-Ampere measure of the Ronkin function has interesting properties which we explore. This measure can be used to derive an upper bound on the area of an amoeba in two dimensions. We also obtain results on the number of complement components of an amoeba, and consider possible extensions of the theory to varieties of codimension higher than 1.

Part II: Differential equations in the complex plane

We consider polynomials in one complex variable arising as eigenfunctions of certain differential operators, and obtain results on the distribution of their zeros. We show that in the limit when the degree of the polynomial approaches innity, its zeros are distributed according to a certain probability measure. This measure has its support on the union of nitely many curve segments, and can be characterized by a simple condition on its Cauchy transform.

Part III: Radon transforms and tomography

This part is concerned with different weighted Radon transforms in two dimensions, in particular the problem of inverting such transforms. We obtain stability results of this inverse problem for rather general classes of weights, including weights of attenuation type with data acquisition limited to a 180 degrees range of angles. We also derive an inversion formula for the exponential Radon transform, with the same restriction on the angle.

Place, publisher, year, edition, pages
Matematiska institutionen, 2003. 123 p.
Laurent series, Harnack curves, differential equations, tomography
National Category
Algebra and Logic
urn:nbn:se:su:diva-15 (URN)91-7265-738-3 (ISBN)
Public defence
2003-10-10, sal 14, hus 5, Kräftriket, Stockholm, 10:00
Available from: 2003-10-02 Created: 2003-10-02Bibliographically approved

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