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- CLAUS SCHEIDERER
- 1999

Let V be an affine algebraic variety over R (or any other real closed field R). We ask when it is true that every positive semidefinite (psd) polynomial function on V is a sum of squares (sos). We show that for dimV ≥ 3 the answer is always negative if V has a real point. Also, if V is a smooth non-rational curve all of whose points at infinity are real,… (More)

- CLAUS SCHEIDERER
- 2002

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)

- Claus Scheiderer
- J. Complexity
- 2005

- CLAUS SCHEIDERER
- 2004

Let g1, . . . , gr ∈ R[x1, . . . , xn] such that the set K = {g1 ≥ 0, . . . , gr ≥ 0} in Rn is compact. We study the problem of representing polynomials f with f |K ≥ 0 in the form f = s0 + s1g1 + · · · + srgr with sums of squares si, with particular emphasis on the case where f has zeros in K. Assuming that the quadratic module of all such sums is… (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)

- CLAUS SCHEIDERER
- 2004

Consider real polynomials g1, . . . , gr in n variables, and assume that the subset K = {g1 ≥ 0, . . . , gr ≥ 0} of Rn is compact. We show that a polynomial f has a representation

- CLAUS SCHEIDERER
- 2001

Let A be a semilocal ring. We compare the set of positive semidefinite (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)

- CLAUS SCHEIDERER
- 2012

We prove that the closed convex hull of any one-dimensional semialgebraic subset of Rn has a semidefinite representation, meaning that it can be written as a linear projection of the solution set of some linear matrix inequality. This is proved by an application of the moment relaxation method. Given a nonsingular affine real algebraic curve C and a compact… (More)

- CLAUS SCHEIDERER
- 2013

We construct families of explicit polynomials f over Q that are sums of squares of polynomials over R, but not over Q. Whether or not such examples exist was an open question originally raised by Sturmfels. We also study representations of f as sums of squares of rational functions over Q. In the case of ternary quartics, we prove that our counterexamples… (More)