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Topics in geometry, analysis and inverse problems
Stockholm University, Faculty of Science, Department of Mathematics.
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.
##### Keyword [en]
Laurent series, Harnack curves, differential equations, tomography
##### National Category
Algebra and Logic
##### Identifiers
ISBN: 91-7265-738-3OAI: oai:DiVA.org:su-15DiVA: diva2:190169
##### Public defence
2003-10-10, sal 14, hus 5, Kräftriket, Stockholm, 10:00
##### Supervisors
Available from: 2003-10-02 Created: 2003-10-02Bibliographically approved
##### List of papers
1. Stratication des espaces de polynômes de Laurent et structure de leurs amibes.
Open this publication in new window or tab >>Stratication des espaces de polynômes de Laurent et structure de leurs amibes.
2000 (English)In: Comptes rendus de l'Académie des sciences. Série 1, Mathématique, ISSN 0764-4442, Vol. 331, no 5, 355-358 p.Article in journal (Refereed) Published
##### Abstract [en]

We consider the stratification of spaces of Laurent polynomials according to the presence of connected components in the complement of the amoeba. Some results concerning the qualitative description of this stratification are obtained.

##### Identifiers
urn:nbn:se:su:diva-23101 (URN)10.1016/S0764-4442(00)01613-X (DOI)
##### Note
Part of urn:nbn:se:su:diva-15Available from: 2003-10-02 Created: 2003-10-02 Last updated: 2009-10-07Bibliographically approved
2. Amoebas, Monge-Ampère measures, and triangulations of the Newton polytope.
Open this publication in new window or tab >>Amoebas, Monge-Ampère measures, and triangulations of the Newton polytope.
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.

##### Identifiers
urn:nbn:se:su:diva-23102 (URN)
##### Note
Part of urn:nbn:se:su:diva-15Available from: 2003-10-02 Created: 2003-10-02 Last updated: 2009-10-07Bibliographically approved
3. Amoebas of maximal area.
Open this publication in new window or tab >>Amoebas of maximal area.
2001 (English)In: International mathematics research notices, ISSN 1073-7928, Vol. 2002, no 9, 441-451 p.Article in journal (Refereed) Published
##### Abstract [en]

To any algebraic curve A in (*)2 one may associate a closed infinite region A in 2 called the amoeba of A. The amoebas of different curves of the same degree come in different shapes and sizes. All amoebas in (*)2 have finite area and, furthermore, there is an upper bound on the area in terms of the degree of the curve. The subject of this paper is the curves in (*)2 whose amoebas are of the maximal area. We show that up to multiplication by a constant in (*)2, such curves are defined over and, furthermore, that their real loci are isotopic to so-called Harnack curves.

##### Identifiers
urn:nbn:se:su:diva-23103 (URN)10.1155/S107379280100023X (DOI)
##### Note
Part of urn:nbn:se:su:diva-15Available from: 2003-10-02 Created: 2003-10-02 Last updated: 2009-10-07Bibliographically approved
4. On polynomial eigenfunctions for a class of differential operators.
Open this publication in new window or tab >>On polynomial eigenfunctions for a class of differential operators.
(English)In: Mathematical Research Letters, ISSN 1073-2780Article in journal (Refereed) Published
##### Identifiers
urn:nbn:se:su:diva-23104 (URN)
##### Note
Part of urn:nbn:se:su:diva-15Available from: 2003-10-02 Created: 2003-10-02 Last updated: 2009-10-07Bibliographically approved
5. An explicit inversion formula for the exponential Radon transform using data from 180˚
Open this publication in new window or tab >>An explicit inversion formula for the exponential Radon transform using data from 180˚
2004 (English)In: Arkiv för matematik, ISSN 0004-2080, Vol. 42, no 2, 253-262 p.Article in journal (Refereed) Published
##### Abstract [en]

We derive a direct inversion formula for the exponential Radon transform. Our formula requires only the values of the transform over an 180° range of angles. It is an explicit formula, except that it involves a holomorphic function for which an explicit expression has not been found. In practice, this function can be approximated by an easily computed polynomial of rather low degree.

Mathematics
##### Identifiers
urn:nbn:se:su:diva-23105 (URN)10.1007/BF02385485 (DOI)
##### Note
Part of urn:nbn:se:su:diva-15Available from: 2003-10-02 Created: 2003-10-02 Last updated: 2009-10-21Bibliographically approved

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##### By organisation
Department of Mathematics
##### On the subject
Algebra and Logic