Claus Scheiderer

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Given an affine algebraic variety V over R with compact set V (R) of real points, and a non-negative polynomial function f ∈ R[V ] with finitely many real zeros, we establish a local-global criterion for f to be a sum of squares in R[V ]. We then specialize to the case where V is a curve. The notion of virtual compactness is introduced, and it is shown that(More)
Consider real polynomials g 1 ,. .. , gr in n variables, and assume that the subset K = {g 1 ≥ 0,. .. , gr ≥ 0} of R n is compact. We show that a polynomial f has a representation f = e∈{0,1} r se · g e 1 1 · · · g er r (*) in which the se are sums of squares, if and only if the same is true in every localization of the polynomial ring by a maximal ideal.(More)
Hilbert proved that a non-negative real quartic form f (x, y, z) is the sum of three squares of quadratic forms. We give a new proof which shows that if the plane curve Q defined by f is smooth, then f has exactly 8 such representations, up to equivalence. They correspond to those real 2-torsion points of the Jacobian of Q which are not represented by a(More)
Given two positive definite forms f, g ∈ R[x 0 ,. .. , xn], we prove that f g N is a sum of squares of forms for all sufficiently large N ≥ 0. We generalize this result to projective R-varieties X as follows. Suppose that X is reduced without one-dimensional irreducible components, and X(R) is Zariski dense in X. Given everywhere positive global sections f(More)
Let A be a semilocal ring. We compare the set of positive semidef-inite (psd) elements of A and the set of sums of squares in A. For psd f ∈ A, whether f is a sum of squares or not depends only on the behavior of f in an infinitesimal neighborhood of the real zeros of f in Spec A. We apply this observation, first to 1-dimensional local rings, then to(More)
A theorem of Grothendieck asserts that over a perfect eld k of cohomological dimension one, all non-abelian H 2-cohomology sets of algebraic groups are trivial. The purpose of this paper is to establish a formally real generalization of this theorem. The generalization | to the context of perfect elds of virtual cohomological dimension one | takes the form(More)