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

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.Keywords: Math, Maths, Mathematic, Mathematics, Mathematical, Pure Mathematics, Theoretical Mathematics, Applied Mathematics, Algorithm, Algorithms, Roots, Critical Points, Complex Analysis, Complex Polynomials, Real Algebra, Real Polynomials, Real Closed Field, Real Radical Ideal, Real Radical Ideal Membership, Ideal, Ideal Membership, Groebner, Gr ̈obner, Groebner Fan, Gr ̈obner Fan, Certificate, Certificate of Real Radical Ideal Membership, Certificate of Ideal Membership, Buchberger’s algorithm, S-Procedure, Real Algebraic Geometry, Algebraic Geometry, Linear Matrix Inequality, Linear Matrix Inequalities, LMI, Cylindrical Algebraic Decomposition, Procedure, Resultant, Lifting, Elimination, Input, Output, Semidefinite, Semidefinite Set, Semidefinite Sets, Semidefinite Programming, Nonlinear Programming, SDP, Relaxation Order, Semidefinite Relaxation, Slack Variable, Linear Operator, Linear Algebra, Moment, Moments, Moment Theory, Optimization, Optimum, Maximum, Supremum, Infimum, Minimum, Optimal, Maximal, Minimal, Extremal Polynomials, Convex Optimization, Convex Analysis, Functional Analysis, Function, Functional, Measure, Positivity, Positivity Certificate, Certificate, Polynomial, Positive Polynomial, Positive Polynomials, Sums of Squares, Ring, Preordering, Quadratic Module, Anneaux preordonnes, Rational Sums of Squares, SOS, Hilbert, David Hilbert, Artin, Emil Artin, Nullstellensatz, Positivestellensatz, Nichtnegativstellensatz, Nonnegative, Non-negative, Nonnegative Polynomial, Nonnegative Polynomials, Non-negative Polynomials, Tarski, Transfer Principle, Tarski-Transfer, Tarski-Seidenberg, Tarski-Seidenberg Principle, Dynamics, Nonlinear Dynamics, Fundamental Theorem of Algebra, Cauchy Integral Formula, The Argument Principle, Liouville’s Theorem, Rouche’s Theorem, Asymptotic, Asymptotics, Derivative, Differential Operator, Pullback, Complexity, Double Exponential, Computational Complexity, Numerics, Numerical, Decision, Decision Theory, Decision Problem, Decision Problems, Decidable, Axiom, Axiomatic, Independence, Model Theory, Higher-Order Logic, Conjecture, Blagovest Sendov, Sendov’s Conjecture, Ilief’s Conjecture, Duality, Duality Gap, Moment and Positive Polynomial Duality, Machine Error, Machine Precision, Validation, True, Truth, Incomplete, Incompleteness, Complete, Completeness, Contrapositive, Consistent, Formal Language, Language, Bounds, Global Optimum, Global Optimization, Constrained Optimizatio