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Positive Polynomials From Hilbert"s 17th Problem to Real Algebra (Springer Monographs in Mathematics) by Alexander Prestel

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  • 21 Currently reading

Published by Springer .
Written in English

Subjects:

  • Algebra,
  • Polynomials,
  • Algebraic Topology,
  • Mathematics,
  • Science/Mathematics,
  • Mathematical Analysis,
  • Mathematics / Algebra / General,
  • Finite Mathematics,
  • Geometry - Algebraic

Book details:

The Physical Object
FormatHardcover
Number of Pages268
ID Numbers
Open LibraryOL9057109M
ISBN 103540412158
ISBN 109783540412151

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In many areas of mathematics (like analysis, real algebraic geometry, functional analysis, etc.) it shows up as positivity of a polynomial on a certain subset of R^n which itself is often given by polynomial inequalities. The main objective of the book is to give useful characterizations of such polynomials.   This book provides an elementary introduction to positive polynomials and sums of squares, the relationship to the moment problem, and the application to polynomial optimization. The focus is on the exciting new developments that have taken place in the last 15 years, arising out of Schmudgen's solution to the moment problem in the compact case Cited by: This book provides an elementary introduction to positive polynomials and sums of squares, the relationship to the moment problem, and the application to polynomial optimization. The focus is on the exciting new developments that have taken place in the last 15 years, arising out of Schmüdgen's solution to the moment problem in the compact.   Positivity is one of the most basic mathematical concepts. In many areas of mathematics (like analysis, real algebraic geometry, functional analysis, etc.) it shows up as positivity of a polynomial on a certain subset of R^n which itself is often given by polynomial inequalities. The main objective of the book is to give useful characterizations of such polynomials.

Positive Polynomials in Control originates from an invited session presented at the IEEE CDC and gives a comprehensive overview of existing results in this quickly emerging area. This carefully edited book collects important contributions from several fields of control, optimization, and mathematics, in order to show different views and approaches of polynomial positivity. Positive Semidefinite Matrices Chapter 1. Positive Polynomials and Sums of Squares Preliminaries on Polynomials Positive Polynomials Extending Positive Polynomials Hilbert's 17th Problem Baer-Krull Theorem Formal Power Series Rings Chapter 2. Krivine's Positivstellensatz Quadratic Modules and Preorderings. Thus, efficient optimization methods can be employed to solve diverse problems involving polynomials. This book gathers the main recent results on positive trigonometric polynomials within a unitary framework; the theoretical results are obtained partly from the general theory of real polynomials, partly from self-sustained developments. the solution offered a “global” characterization of positivity of polynomials on R(n). This brings us to the main goal of our book: we seek characterizations of those polynomials fthat are positive on certain sets, themselves defined by polynomial inequalities. In every case, these characterizations consist of.

Many important applications in global optimization, algebra, probability and statistics, applied mathematics, control theory, financial mathematics, inverse problems, etc. can be modeled as a particular instance of the Generalized Moment Problem (GMP).This book introduces a new general methodology to solve the GMP when its data are polynomials and basic semi-algebraic : $ This book introduces a new general methodology to solve the GMP when its data are polynomials and basic semi-algebraic sets. This methodology combines semidefinite programming with recent results from real algebraic geometry to provide a hierarchy of semidefinite . Section snippets Positivity and stability of polynomial matrices. Let D = s∈ C: 1 s ★ a b b ★ c H 1 s positive eigenvalue and one negative eigenvalue. Standard choices for D are the left half-plane (a=0,b=1,c=0) and the unit disk (a=−1,b=0,c=1). included: There are the books by Prestel and Delzell [69] and Marshall [44] on positive polynomials, a survey article by Reznick [75] about psd and sos polynomials with a wealth of historical information, and a recent survey article by Scheiderer [84] on positivity .