An elementary and constructive solution to Hilbert's 17th Problem for matrices

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dc.creatorHillar, Christopher J.
dc.creatorNie, Jiawang
dc.date2006-10-12
dc.date2006-10-27
dc.date.accessioned2026-07-07T07:28:59Z
dc.date.available2026-07-07T07:28:59Z
dc.descriptionWe give a short and elementary proof of a theorem of Procesi, Schacher and (independently) Gondard, Ribenboim that generalizes a famous result of Artin. Let $A$ be an $n \times n$ symmetric matrix with entries in the polynomial ring $\mathbb R[x_1,...,x_m]$. The result is that if $A$ is postive semidefinite for all substitutions $(x_1,...,x_m) \in \mathbb R^m$, then $A$ can be expressed as a sum of squares of symmetric matrices with entries in $\mathbb R(x_1,...,x_m)$. Moreover, our proof is constructive and gives explicit representations modulo the scalar case.
dc.description3 pages, generalized and added 2 examples
dc.identifierhttps://arxiv.org/abs/math/0610388
dc.identifierhttp://arxiv.org/abs/math/0610388
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/117937
dc.subjectRings and Algebras
dc.subjectAlgebraic Geometry
dc.subject12D15, 03C64, 13L05, 14P05, 15A21, 15A54
dc.titleAn elementary and constructive solution to Hilbert's 17th Problem for matrices
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