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Medientyp:
E-Artikel
Titel:
Positive polynomials and sequential closures of quadratic modules
Beteiligte:
Netzer, Tim
Erschienen:
American Mathematical Society (AMS), 2009
Erschienen in:
Transactions of the American Mathematical Society, 362 (2009) 5, Seite 2619-2639
Sprache:
Englisch
DOI:
10.1090/s0002-9947-09-05001-6
ISSN:
0002-9947;
1088-6850
Entstehung:
Anmerkungen:
Beschreibung:
Let S = { x ∈ R n ∣ f 1 ( x ) ≥ 0 , … , f s ( x ) ≥ 0 } \mathcal {S}=\{x\in \mathbb {R}^n\mid f_1(x)\geq 0,\ldots ,f_s(x)\geq 0\} be a basic closed semi-algebraic set in R n \mathbb {R}^n and let P O ( f 1 , … , f s ) \mathrm {PO}(f_1,\ldots ,f_s) be the corresponding preordering in R [ X 1 , … , X n ] \mathbb {R}[X_1,\ldots ,X_n] . We examine for which polynomials f f there exist identities \[ f + ε q ∈ P O ( f 1 , … , f s ) for all ε > 0. f+\varepsilon q\in \mathrm {PO}(f_1,\ldots ,f_s) \mbox { for all } \varepsilon >0. \] These are precisely the elements of the sequential closure of P O ( f 1 , … , f s ) \mathrm {PO}(f_1,\ldots ,f_s) with respect to the finest locally convex topology. We solve the open problem from Kuhlmann, Marshall, and Schwartz (2002, 2005), whether this equals the double dual cone \[ P O ( f 1 , … , f s ) ∨ ∨ , \mathrm {PO}(f_1,\ldots ,f_s)^{\vee \vee }, \] by providing a counterexample. We then prove a theorem that allows us to obtain identities for polynomials as above, by looking at a family of fibre-preorderings, constructed from bounded polynomials. These fibre-preorderings are easier to deal with than the original preordering in general. For a large class of examples we are thus able to show that either every polynomial f f that is nonnegative on S \mathcal {S} admits such representations, or at least the polynomials from P O ( f 1 , … , f s ) ∨ ∨ \mathrm {PO}(f_1,\ldots ,f_s)^{\vee \vee } do. The results also hold in the more general setup of arbitrary commutative algebras and quadratic modules instead of preorderings.