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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|Commitee:||Stopple, Jeffrey, Bamieh, Bassam|
|School:||University of California, Santa Barbara|
|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|
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