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Softcover ISBN:  9781470441081 
Product Code:  MEMO/263/1277 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $51.00 
eBook ISBN:  9781470456627 
Product Code:  MEMO/263/1277.E 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $51.00 
Softcover ISBN:  9781470441081 
eBook ISBN:  9781470456627 
Product Code:  MEMO/263/1277.B 
List Price:  $170.00 $127.50 
MAA Member Price:  $153.00 $114.75 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 263; 2020; 125 ppMSC: Primary 12; 14; 13
The authors prove an elementary recursive bound on the degrees for Hilbert's 17th problem. More precisely they express a nonnegative polynomial as a sum of squares of rational functions and obtain as degree estimates for the numerators and denominators the following tower of five exponentials \[ 2^{ 2^{ 2^{d^{4^{k}}} } } \] where \(d\) is the number of variables of the input polynomial. The authors' method is based on the proof of an elementary recursive bound on the degrees for Stengle's Positivstellensatz. More precisely the authors give an algebraic certificate of the emptyness of the realization of a system of sign conditions and obtain as degree bounds for this certificate a tower of five exponentials, namely \[ 2^{ 2^{\left(2^{\max\{2,d\}^{4^{k}}}+ s^{2^{k}}\max\{2, d\}^{16^{k}{\mathrm bit}(d)} \right)} } \] 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.

Table of Contents

Chapters

1. Introduction

2. Weak inference and weak existence

3. Intermediate value theorem

4. Fundamental theorem of algebra

5. Hermite’s Theory

6. Elimination of one variable

7. Proof of the main theorems

8. Annex


Additional Material

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The authors prove an elementary recursive bound on the degrees for Hilbert's 17th problem. More precisely they express a nonnegative polynomial as a sum of squares of rational functions and obtain as degree estimates for the numerators and denominators the following tower of five exponentials \[ 2^{ 2^{ 2^{d^{4^{k}}} } } \] where \(d\) is the number of variables of the input polynomial. The authors' method is based on the proof of an elementary recursive bound on the degrees for Stengle's Positivstellensatz. More precisely the authors give an algebraic certificate of the emptyness of the realization of a system of sign conditions and obtain as degree bounds for this certificate a tower of five exponentials, namely \[ 2^{ 2^{\left(2^{\max\{2,d\}^{4^{k}}}+ s^{2^{k}}\max\{2, d\}^{16^{k}{\mathrm bit}(d)} \right)} } \] 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.

Chapters

1. Introduction

2. Weak inference and weak existence

3. Intermediate value theorem

4. Fundamental theorem of algebra

5. Hermite’s Theory

6. Elimination of one variable

7. Proof of the main theorems

8. Annex