| Description |
1 online resource (138 pages) |
| Series |
Memoirs of the American Mathematical Society Ser. ; v. 263 |
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Memoirs of the American Mathematical Society Ser
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| Contents |
Cover -- Title page -- Chapter 1. Introduction -- 1.1. Hilbert's 17th problem -- 1.2. Positivstellensatz -- 1.3. Historical background on constructive proofs and degree bounds -- 1.4. Our results -- 1.5. Organization of the paper -- Acknowledgements -- Chapter 2. Weak inference and weak existence -- 2.1. Weak inference -- 2.2. Weak existence -- 2.3. Complex numbers -- 2.4. Identical polynomials -- 2.5. Matrices -- Chapter 3. Intermediate value theorem -- 3.1. Intermediate value theorem -- 3.2. Real root of a polynomial of odd degree -- Chapter 4. Fundamental theorem of algebra |
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4.1. Fundamental theorem of algebra -- 4.2. Decomposition of a polynomial into irreducible real factors -- 4.3. Decomposition of a polynomial into irreducible real factors with multiplicities -- Chapter 5. Hermite's Theory -- 5.1. Signature of Hermite's quadratic form and real root counting -- 5.2. Signature of Hermite's quadratic form and signs of principal minors -- 5.3. Sylvester Inertia Law -- 5.4. Hermite's quadratic form and Sylvester Inertia Law -- Chapter 6. Elimination of one variable -- 6.1. Thom encoding of real algebraic numbers |
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6.2. Conditions on the parameters fixing the Thom encoding -- 6.3. Conditions on the parameters fixing the real root order on a family -- 6.4. Realizable sign conditions on a family of polynomials -- Chapter 7. Proof of the main theorems -- Chapter 8. Annex -- Bibliography -- Back Cover |
| Summary |
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 2̂{ 2̂{ 2̂{d̂{4̂{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 |
| Notes |
Print version record |
| Subject |
Polynomials.
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Algebraic fields.
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Recursive functions.
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Algebraic stacks.
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Algebraic stacks
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Algebraic fields
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Polynomials
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Recursive functions
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| Form |
Electronic book
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| Author |
Perrucci, Daniel
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Roy, Marie-Françoise
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| ISBN |
9781470456627 |
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1470456621 |
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