Author | Prestel, Alexander. author |
---|---|

Title | Positive Polynomials [electronic resource] : From Hilbert's 17th Problem to Real Algebra / by Alexander Prestel, Charles N. Delzell |

Imprint | Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 2001 |

Connect to | http://dx.doi.org/10.1007/978-3-662-04648-7 |

Descript | VIII, 269 p. online resource |

SUMMARY

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 R̂n which itself is often given by polynomial inequalities. The main objective of the book is to give useful characterizations of such polynomials. It takes as starting point Hilbert's 17th Problem from 1900 and explains how E. Artin's solution of that problem eventually led to the development of real algebra towards the end of the 20th century. Beyond basic knowledge in algebra, only valuation theory as explained in the appendix is needed. Thus the monograph can also serve as the basis for a 2-semester course in real algebra

CONTENT

1. Real Fields -- 2. Semialgebraic Sets -- 3. Quadratic Forms over Real Fields -- 4. Real Rings -- 5. Archimedean Rings -- 6. Positive Polynomials on Semialgebraic Sets -- 7. Sums of 2mth Powers -- 8. Bounds -- Appendix: Valued Fields -- A.1 Valuations -- A.2 Algebraic Extensions -- A.3 Henselian Fields -- A.4 Complete Fields -- A.5 Dependence and Composition of Valuations -- A.6 Transcendental Extensions -- A.7 Exercises -- A.8 Bibliographical Comments -- References -- Glossary of Notations

Mathematics
Algebra
Algebraic geometry
Functional analysis
Mathematics
Algebra
Algebraic Geometry
Functional Analysis