Dissertation/Thesis Abstract

Roots and Critical Points of Complex Polynomials: Applications of Algorithms in Real Algebra, Moment Theory, Convex Analysis, Optimization, and Positive Polynomials to a Conjecture in Pure Mathematics
by Spjut, Richard William, Ph.D., University of California, Santa Barbara, 2020, 185; 27997138
Abstract (Summary)

Conjecture 0.1 (Conjecture of Blagovest Sendov (1958)):

For a complex polynomial of degree two or more with all its roots contained within the closed unit disk, each root has a critical point within unit distance.

We introduce a countable collection of conjectures – one for each degree – by transferring into languages of Real Algebra. For fixed degree, each conjecture is decidable. Thus, we consider decidable statements in Real Algebra. For each of these decidable statements, we seek for certificates. We provide variations of this theme for different contexts: Positivstellensatz, Nichtnegativstellensatz and real radical ideal membership. In our context, our positivity certificates (or membership certificates) provide proof when achieved. We also find plenty of novel numerical evidence substantiating our conjectures.

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Indexing (document details)
Advisor: Putinar, Mihai
Commitee: Stopple, Jeffrey, Bamieh, Bassam
School: University of California, Santa Barbara
Department: Mathematics
School Location: United States -- California
Source: DAI-B 82/3(E), Dissertation Abstracts International
Subjects: Mathematics, Theoretical Mathematics, Applied Mathematics
Keywords: Blagovest Sendov Conjecture, Complex Analysis, Optimization, Real Algebra, Real Radical Ideal, Semidefinite Programming
Publication Number: 27997138
ISBN: 9798672103808
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