Abstract
In this chapter we shall show how to obtain more information about representations of polynomials that are positive on semialgebraic sets. We shall deal with representations obtained in Theorem 3.5.8 using squares of rational functions, and in Theorem 5.2.9 using squares of polynomials. We shall also deal with the distinguished representations of Theorem 6.3.4. In all cases we shall obtain effective bounds on the degree of the (sums of) squares used in the representation. The bound will be computable from certain invariants attached to the polynomials h 1,..., h s defining the semialgebraic set W ℝ(h), and the polynomial f that is (strictly) positive on W ℝ(h). Not surprisingly, the degrees of f and the h i’s are among such invariants. This, however, in general will not be sufficient. What else is needed will be a subject of our search.
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© 2001 Springer-Verlag Berlin Heidelberg
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Prestel, A., Delzell, C.N. (2001). Bounds. In: Positive Polynomials. Springer Monographs in Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-04648-7_9
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DOI: https://doi.org/10.1007/978-3-662-04648-7_9
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-07445-5
Online ISBN: 978-3-662-04648-7
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