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- Jaka Cimprič, Salma Kuhlmann, Claus Scheiderer
- 2008

We begin a systematic study of positivity and moment problems in an equivariant setting. Given a reductive group G over R acting on an affine R-variety V , we consider the induced dual action on the coordinate ring R[V ] and on the linear dual space of R[V ]. In this setting, given an invariant closed semialgebraic subset K of V (R), we study the problem of… (More)

- Claus Scheiderer
- 2011

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)

- Albrecht Pfister, Claus Scheiderer
- 2010

In 1888, Hilbert proved that every nonnegative quartic form f = f (x, y, z) with real coefficients is a sum of three squares of quadratic forms. His proof was ahead of its time and used advanced methods from topology and algebraic geometry. Up to now, no elementary proof is known. Here we present a completely new approach. Although our proof is not easy, it… (More)

In recent years, much work has been devoted to a systematic study of polynomial identities certifying strict or non-strict posi-tivity of a polynomial f on a basic closed set K ⊂ R n. The interest in such identities originates not least from their importance in polynomial optimization. The majority of the important results requires the archimedean… (More)

Continuing our recent work in [5] we study polynomial masks of multivariate tight wavelet frames from two additional and complementary points of view: convexity and system theory. We consider such polynomial masks that are derived by means of the unitary extension principle from a single polynomial. We show that the set of such poly-nomials is convex and… (More)

- Claus Scheiderer
- 2012

We prove that the closed convex hull of any one-dimensional semi-algebraic subset of R n 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… (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)

- David Harari, Claus Scheiderer, Tam´as Szamuely
- 2014

We study local-global questions for Galois cohomology over the function field of a curve defined over a p-adic field, the main focus being weak approximation of rational points. We construct a 9-term Poitou–Tate type exact sequence for tori over a field as above (and also a 12-term sequence for finite modules). Like in the number field case, part of the… (More)