Murray Marshall

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Natural sufficient conditions for a polynomial to have a local minimum at a point are considered. These conditions tend to hold with probability 1. It is shown that polynomials satisfying these conditions at each minimum point have nice presentations in terms of sums of squares. Applications are given to optimization on a compact set and also to global(More)
The paper is a continuation of work initiated by the first two authors in [K–M]. Section 1 is introductory. In Section 2 we give new proofs of results of Scheiderer in [S1] [S2] in the compact case; see Corollaries 2.3, 2.4 and 2.5. The main tool in Section 2, Lemma 2.1, is also used in Section 3 where we continue the examination of the case n = 1 initiated(More)
We make use of a result of Hurwitz and Reznick [8] [19], and a consequence of this result due to Fidalgo and Kovacec [5], to determine a new sufficient condition for a polynomial f ∈ R[X 1 ,. .. , X n ] of even degree to be a sum of squares. This result generalizes a result of Lasserre in [10] and a result of Fidalgo and Kovacec in [5], and it also(More)
We prove that if f (x, y) is a polynomial with real coefficients which is non-negative on the strip [0, 1] × R, then f (x, y) has a presentation of the form f (x, y) = k i=1 g i (x, y) 2 + j=1 h j (x, y) 2 x(1 − x), where the g i (x, y) and h j (x, y) are polynomials with real coefficients.
real spectra [7, Chs. 6–8], also called spaces of signs [1, Ch. 3], arise naturally in the study of semialgebraic sets, more generally, in the study of constructible sets in the real spectrum of a commutative ring with 1. Let A denote the ring of all polynomial functions on V , where V ⊆ R n is an algebraic set. Consider f, g ∈ A to be equivalent if f and g(More)
Let F be an ordered eld, v the unique nest valuation on F compatible with the ordering (so v(a) v(b) ii njbj jaj for some integer n 1), V the value group of v and the residue eld of v (so is archimedean) 5] 10] 19]. If F 0 is an ordered extension of F , then the nest valuation on F 0 compatible with the extended ordering is an extension of v which we denote(More)
Gewidmet meinen Eltern Marina und Michael Plaumann und meinem Onkel Peter Plaumann in tiefer Dankbarkeit Acknowledgements I would like to express my gratitude to my advisor Claus Scheiderer for the many ideas and insights into the topic that he shared with me, as well as for his constant encouragement. I am grateful for many interesting and helpful(More)
Spaces of orderings were introduced by the second author in a series of papers 15]{{19] and various structure results were obtained generalizing results proved earlier for formally real elds by various people: In 6] and 26] (also see 20] and 23] 25]) Craven and Tschimmel prove that formally real skew elds give rise to spaces of orderings. In 8] (also see(More)
We determine new sufficient conditions in terms of the coefficients for a polynomial f ∈ R[X] of degree 2d (d ≥ 1) in n ≥ 1 variables to be a sum of squares of polynomials, thereby strengthening results of Fidalgo and Kovacec [2] and of Lasserre [6]. Exploiting these results, we determine, for any polynomial f ∈ R[X] of degree 2d whose highest degree term(More)