arXiv subjects

Claus Scheiderer

Publications and source records attributed to Claus Scheiderer.

32 records · Page 2Linked to original sources

arXiv2015

Extreme positive ternary sextics

We study nonnegative (psd) real sextic forms $q(x_0,x_1,x_2)$ that are not sums of squares (sos). Such a form has at most ten real zeros. We give a complete and explicit characterization of all sets $S\subset\mathbb{P}^2(\mathbb{R})$ with $|S|=9$ for which there is a psd non-sos sextic vanishing in $S$. Roughly, on every plane cubic $X$ with only real nodes there is a certain natural divisor class $τ_X$ of degree~$9$, and $S$ is the real zero set of some psd non-sos sextic if, and only if, there is a unique cubic $X$ through $S$ and $S$ represents the class $τ_X$ on $X$. If this is the case, there is a unique extreme ray $\mathbb{R}_+q_S$ of psd non-sos sextics through $S$, and we show how to find $q_S$ explicitly. The sextic $q_S$ has a tenth real zero which for generic $S$ does not lie in $S$, but which may degenerate into a higher singularity contained in $S$. We also show that for any eight points in $\mathbb{P}^2(\mathbb{R})$ in general position there exists a psd sextic that is not a sum of squares and vanishes in the given points.

math.AG↗

arXiv2014

An Algebraic Perspective on Multivariate Tight Wavelet Frames. II

Continuing our recent work 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 polynomials is convex and reveal its extremal points as polynomials that satisfy the quadrature mirror filter condition. Multiplicative structure of such polynomial sets allows us to improve the known upper bounds on the number of frame generators derived from box splines. In the univariate and bivariate settings, the polynomial masks of a tight wavelet frame can be interpreted as the transfer function of a conservative multivariate linear system. Recent advances in system theory enable us to develop a more effective method for tight frame constructions. Employing an example by S. W. Drury, we show that for dimension greater than 2 such transfer function representations of the corresponding polynomial masks do not always exist. However, for wavelet masks derived from multivariate polynomials with non-negative coefficients, we determine explicit transfer function representations. We illustrate our results with several examples.

math.FA↗

arXiv2014

Weak approximation for tori over $p$-adic function fields

This is the companion piece to "Local-global questions for tori over p-adic function fields" by the first and third authors. We study local-global questions for Galois cohomology over the function field of a curve defined over a p-adic field, the main focus here 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 sequence can then be used to analyze the defect of weak approximation for a torus. We also show that the defect of weak approximation is controlled by a certain subgroup of the third unramified cohomology group of the torus.

math.NT↗

arXiv2013

Sums of squares of polynomials with rational coefficients

We construct families of explicit polynomials f with rational coefficients that are sums of squares of polynomials over the real numbers, but not over the rational numbers. 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 with rational coefficients. In the case of ternary quartics, we prove that our counterexamples to Sturmfels' question are the only ones.

math.AG↗

arXiv2013

Quillen property of real algebraic varieties

Let I be a conjugation-invariant ideal in the complex polynomial ring with variables z_1,...,z_n and their conjugates. The ideal I has the Quillen property if every real valued, strictly positive polynomial on the real zero set of I in C^n is a sum of hermitian squares modulo I. We first relate the Quillen property to the archimedean property from real algebra. Using hereditary calculus, we then quantize and show that the Quillen property implies the subnormality of commuting tuples of Hilbert space operators satisfying the identities in I. In the finite rank case we give a complete geometric characterization of when the identities in $I$ imply normality for a commuting tuple of matrices. This geometric interpretation provides simple means to refute Quillen's property of an ideal. We also generalize these notions and results from real algebraic sets to semi-algebraic sets in C^n.

math.AG↗

arXiv2012

A real algebra perspective on multivariate tight wavelet frames

Recent results from real algebraic geometry and the theory of polynomial optimization are related in a new framework to the existence question of multivariate tight wavelet frames whose generators have at least one vanishing moment. Namely, several equivalent formulations of the so-called Unitary Extension Principle by Ron and Shen are interpreted in terms of hermitian sums of squares of certain nonnegative trigonometric polynomials and in terms of semi-definite programming. The latter together with the recent results in algebraic geometry and semi-definite programming allow us to answer affirmatively the long standing open question of the existence of such tight wavelet frames in dimension $d=2$; we also provide numerically efficient methods for checking their existence and actual construction in any dimension. We exhibit a class of counterexamples in dimension $d=3$ showing that, in general, the UEP property is not sufficient for the existence of tight wavelet frames. On the other hand we provide stronger sufficient conditions for the existence of tight wavelet frames in dimension $d > 3$ and illustrate our results by several examples.

math.FA↗

arXiv2011

A Positivstellensatz for projective real varieties

Given two positive definite forms f, g in R[x_0,...,x_n], we prove that fg^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 of L^{\otimes2} and g of M^{\otimes2}, where L, M are invertible sheaves on X and M is ample, fg^N is a sum of squares of sections of L\otimes M^{\otimes N} for all large N >= 0. In fact we prove a much more general version with semi-algebraic constraints, defined by sections of invertible sheaves. For nonsingular curves and surfaces and sufficiently regular constraints, the result remains true even if f is just nonnegative. The main tools are local-global principles for sums of squares, and on the other hand an existence theorem for totally real global sections of invertible sheaves, which is the second main result of this paper. For this theorem, X may be quasi-projective, but again should not have curve components. In fact, this result is false for curves in general.

math.AG↗

arXiv2010

An elementary proof of Hilbert's theorem on ternary quartics

In 1888, Hilbert proved that every non-negative 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 uses only elementary techniques. As a by-product, it gives information on the number of representations f=p_1^2+p_2^2+p_3^2 of f up to orthogonal equivalence. We show that this number is 8 for generically chosen f, and that it is 4 when f is chosen generically with a real zero. Although these facts were known, there was no elementary approach to them so far.

math.AG↗

arXiv2010

The ring of bounded polynomials on a semi-algebraic set

Let V be a normal affine variety over the real numbers R, and let S be a semi-algebraic subset of V(R). We study the subring B(S) of the coordinate ring of V consisting of the polynomials that are bounded on S. We introduce the notion of S-compatible completions of V, and we prove the existence of such completions when V is of dimension at most 2 or S=V(R). An S-compatible completion X of V yields an isomorphism of B(S) with the ring of regular functions on some (concretely specified) open subvariety of X. We prove that B(S) is a finitely generated R-algebra if S is open and of dimension at most 2, and we show that this result becomes false in higher dimensions.

math.AG↗

arXiv2010

Convex hulls of curves of genus one

Let C be a real nonsingular affine curve of genus one, embedded in affine n-space, whose set of real points is compact. For any polynomial f which is nonnegative on C(R), we prove that there exist polynomials f_i with f \equiv \sum_i f_i^2 (modulo I_C) and such that the degrees deg(f_i) are bounded in terms of deg(f) only. Using Lasserre's relaxation method, we deduce an explicit representation of the convex hull of C(R) in R^n by a lifted linear matrix inequality. This is the first instance in the literature where such a representation is given for the convex hull of a nonrational variety. The same works for convex hulls of (singular) curves whose normalization is C. We then make a detailed study of the associated degree bounds. These bounds are directly related to size and dimension of the projected matrix pencils. In particular, we prove that these bounds tend to infinity when the curve C degenerates suitably into a singular curve, and we provide explicit lower bounds as well.

math.AG↗

arXiv2009

Pure states, nonnegative polynomials and sums of squares

In recent years, much work has been devoted to a systematic study of polynomial identities certifying strict or non-strict positivity of a polynomial on a basic closed semialgebraic set. The interest in such identities originates not least from their importance in polynomial optimization. The majority of the important results requires the archimedean condition, which implies that the semialgebraic set has to be compact. This paper introduces the technique of pure states into commutative algebra. We show that this technique allows an approach to most of the recent archimedean Stellensaetze that is considerably easier and more conceptual than the previous proofs. In particular, we reprove and strengthen some of the most important results from the last years. In addition, we establish several such results which are entirely new. They are the first that allow the polynomial to have arbitrary, not necessarily discrete, zeros on the semialgebraic set.

math.AC↗

arXiv2008

Sums of squares and moment problems in equivariant situations

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 representation of invariant nonnegative polynomials on $K$ by invariant sums of squares, and the closely related problem of representation of invariant linear functionals on $\R[V]$ by invariant measures supported on $K$. To this end, we analyse the relation between quadratic modules of $\R[V]$ and associated quadratic modules of the (finitely generated) subring $\R[V]^G$ of invariant polynomials. We apply our results to investigate the finite solvability of an equivariant version of the multidimensional $K$-moment problem. Most of our results are specific to the case where the group $G(\R)$ is compact.

math.AG↗

arXiv2004

A New Proof of Hilbert's Theorem on Ternary Quartics

David 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 complex 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 conjugation-invariant divisor on Q.

math.AG↗

arXiv1998

Grothendieck's theorem on non-abelian H^2 and local-global principles

A theorem of Grothendieck asserts that over a perfect field 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 fields of virtual cohomological dimension one -- takes the form of a local-global principle for the H^2-sets with respect to the orderings of the field. This principle asserts in particular that an element in H^2 is neutral precisely when it is neutral in the real closure with respect to every ordering in a dense subset of the real spectrum of k. Our techniques provide a new proof of Grothendieck's original theorem. An application to homogeneous spaces over k is also given.

math.AG↗