Notions of Positivity and the Geometry of Polynomials [electronic resource] /
Brändén, Petter.editor.
Passare, Mikael.editor.
Putinar, Mihai.editor.
The book consists of solicited articles from a select group of mathematicians and physicists working at the interface between positivity and the geometry, combinatorics or analysis of polynomials of one or several variables. It is dedicated to the memory of Julius Borcea (1968-2009), a distinguished mathematician, Professor at the University of Stockholm. With his extremely original contributions and broad vision, his impact on the topics of the planned volume cannot be underestimated. All contributors knew or have exchanged ideas with Dr. Borcea, and their articles reflect, at least partially, his heritage.
Description based upon print version of record.
Notions of Positivity and the Geometry of Polynomials; Contents; Introduction; SECONDE COMPOSITION DE MATHEMATIQUES; Second Composition in Mathematics; I. A few simple cases of the conjecture; II. The case of a real root; III. Continuity of the roots of a polynomial; IV Extremal polynomials; Publications of Julius Borcea; Commuting Linear Differential Operators and Hankel Matrices; 1. Introduction; 2. Hankel operators of finite rank; 3. The case when U'/U is rational; References; Variants of the Effective Nullstellensatz and Residue Calculus; 1. Introduction; 2. Product ideals
3. Division problems and residues, the basic set-up4. Residue calculus; 5. Integral closure and distinguished varieties; 6. Proofs of the theorems; 7. The worst possible situation for the Nullstellensatz; References; An Extension Problem for Convex Functions; 1. Introduction; 2. Proof of Theorem 1; 3. An extremal convex function; References; Subharmonic Configurations and Algebraic Cauchy Transforms of Probability Measures; Introduction; 1. A key lemma; 1.1. Local subharmonic configurations; 1.2. A relaxed assumption; 2. Subharmonic configurations: The general piecewise harmonic case
2.1. A description of ?(V )2.2. Harmonic level sets; 2.3. Local subharmonic configurations; 2.4. A non-transversal case; 3. Piecewise holomorphic functions; 3.1. The non-return lemma; 3.2. Completing the proof of Proposition 3.1; 3.3. Proof of Theorem 1; 4. On algebraic root functions: Proof of Theorem 2; 4.1. A result inside sectors; 4.2. Condition on ?G-derivatives; 4.3. Further preparation for the proof of Theorem 2; 4.4. Tangential derivatives; 4.5. Proof of Theorem 2; 5. Further directions and open problems; 5.1. Existence of measures in m+alg
5.2. Combinatorics of subharmonic configurations5.3. Plurisubharmonic configurations; 5.4. Configurations induced by fundamental solutions; Acknowledgement; References; Proof of the Monotone Column Permanent Conjecture; 1. The monotone column permanent conjecture; The Monotone Column Permanent Conjecture (MCPC); The Multivariate MCP Conjecture (MMCPC); 2. Stable polynomials; 3. Proof of the MMCPC; 3.1. Reduction to Ferrers matrices; 3.2. A more symmetrical problem; 3.3. A differential recurrence relation; 3.4. Finally, proof of the MMCPC; 4. Further results
4.1. Generalization to rectangular matrices.4.2. A cycle-counting generalization; 4.3. Multivariate stable Eulerian polynomials; 4.4. Grace's Apolarity Theorem; 4.5. Permanental inequalities; References; Tracial Nullstellensätze; 1. Introduction; 2. Notation and set-up; 2.1. The free algebra; 2.2. The free *-algebra; 2.3. Evaluations and representations; 3. Amitsur's Nullstellensatz; 4. Directional Nullstellensatz; 4.1. Bergman's Nullstellensatz; 5. Tracial Nullstellensätze; 5.1. More notation; 5.2. Involution-free case; 5.2.1. Image of a polynomial and Lie theory.
5.2.2. Tracial Nullstellensätze.
Includes bibliographical references.
The book consists of solicited articles from a select group of mathematicians and physicists working at the interface between positivity and the geometry, combinatorics or analysis of polynomials of one or several variables. It is dedicated to the memory of Julius Borcea (1968-2009), a distinguished mathematician, Professor at the University of Stockholm. With his extremely original contributions and broad vision, his impact on the topics of the planned volume cannot be underestimated. All contributors knew or have exchanged ideas with Dr. Borcea, and their articles reflect, at least partially, his heritage.
Mathematical analysis.
Analysis (Mathematics).
Analysis.