An elementary recursive bound for effective positivstellensatz and Hilbert's 17th problem / Henri Lombardi, Daniel Perrucci, Marie-Françoise Roy.

Format:

Book

Author/Creator:

Lombardi, Henri, author.

Perrucci, Daniel, author.

Roy, M.-F. (Marie-Françoise), author.

Series:

Memoirs of the American Mathematical Society ; no. 1277.

Memoirs of the American Mathematical Society, 0065-9266 ; number 1277

Language:

English

Subjects (All):

Polynomials.

Algebraic fields.

Recursive functions.

Physical Description:

v, 125 pages ; 26 cm.

Place of Publication:

Providence, RI : American Mathematical Society, 2020.

Summary:

"We prove an elementary recursive bound on the degrees for Hilbert's 17th problem. More precisely we express a nonnegative polynomial as a sum of squares of rational functions, and we obtain as degree estimates for the numerators and denominators the following tower of five exponentials 222d4k where d is the degree and k is the number of variables of the input polynomial. Our method is based on the proof of an elementary recursive bound on the degrees for Stengle's Positivstellensatz. More precisely we give an algebraic certificate of the emptyness of the realization of a system of sign conditions and we obtain as degree bounds for this certificate a tower of five exponentials, namely 2²(2max{2,d}4k+s2kmax{2,d}16kbit(d)) where d is a bound on the degrees, s is the number of polynomials and k is the number of variables of the input polynomials-- Provided by publisher.

Contents:

Weak inference and weak existence

Intermediate value theorem

Fundamental theorem of algebra

Hermite's theory

Elimination of one variable

Proof of the main theorems

Annex.

Notes:

Includes bibliographical references (pages 123-125).

Other Format:

Electronic version: Lombardi, Henri. Elementary recursive bound for effective positivstellensatz and Hilbert's 17th problem.

ISBN:

9781470441081

147044108X

OCLC:

1132241005