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

الأصل · EN

We 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 × n symmetric matrix with entries in the polynomial ring R[x₁,...,xₘ]. The result is that if A is postive semidefinite for all substitutions (x₁,...,xₘ) ∈ Rᵐ, then A can be expressed as a sum of squares of symmetric matrices with entries in R(x₁,...,xₘ). Moreover, our proof is constructive and gives explicit representations modulo the scalar case.