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.Comment: 14 p

Semidefinite representation for convex hulls of real algebraic curves

We show 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 compact semialgebraic subset K of its R-points, the preordering P(K) of all regular functions on C that are nonnegative on K is known to be finitely generated. We prove that P(K) is stable, meaning that uniform degree bounds exist for weighted sum of squares representations of elements of P(K). We also extend this last result to the case where K is only virtually compact. The main technical tool for the proof of stability is the archimedean local-global principle. As a consequence of our results we prove that every convex semialgebraic subset of R^2 has a semidefinite representation.Comment: v2: 19 pp (Section 6 is new); v3: 19 pp (small issues fixed); v4: updated and slightly expande

Come On In. The Water's Fine. An Exploration of Web 2.0 Technology and Its Emerging Impact on Foundation Communications

According to the authors of Come on in. The water's fine. An exploration of Web 2.0 technology and its emerging impact on foundation communications, foundations that have adopted new and still emerging forms of digital communications -- interactive Web sites, blogs, wikis, and social networking applications -- are finding that they offer "opportunities for focused convenings and conversations, lend themselves to interactions with and among grantees, and are an effective story-telling medium." The report's authors, David Brotherton and Cynthia Scheiderer, of Brotherton Strategies, who spent nearly a year exploring how foundations are using new media, add that "electronic communications create an opportunity to connect people who are interested in an issue with each other and the grantees working on the issue."The report also acknowledges that the new technologies raise skepticism and concern among foundations. They include the "worry of losing control over the foundation's message, allowing more staff members to represent the foundation in a more public way, opening the flood gates of grant requests or the headache of a forum gone bad with unwanted or inappropriate posts."Still, the report urges foundations to put aside their worries and make even more forceful use of new media applications and tools. The report argues that whatever is "lost in message control will be more than made up for by the opportunity to engage audiences in new ways, with greater programmatic impact."Acknowledging that adoption of new media tools will require some cultural and operational shifts in foundations, the report offers suggestions from Ernest James Wilson III, dean and Walter Annenberg chair in communication at the University of Southern California, for how to deal with these challenges. He says that for foundations to make the best use of what the technology offers, they should concentrate on three things:Build up the individual "human capital" of their staffs and provide them the competencies they need to operate in the new digital world.Make internal institutional reforms to reward creativity and innovation in using these new media internally and among grantees.Build social networks that span sectors and institutions, to engage in ongoing dialogue among private, public, nonprofits and research stakeholders.As Wilson also says, "All of these steps first require leadership, arguably a new type of leadership, not only at the top but also from the 'bottom' up, since many of the people with the requisite skills, attitudes, substantive knowledge and experience are younger, newer employees, and occupy the low-status end of the organizational pyramid, and hence need strong allies at the top.

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.Comment: 26 page

Weak approximation for tori over pp-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.Comment: final version, to appear in IMR

Toric completions and bounded functions on real algebraic varieties

Given a semi-algebraic set S, we study compactifications of S that arise from embeddings into complete toric varieties. This makes it possible to describe the asymptotic growth of polynomial functions on S in terms of combinatorial data. We extend our earlier work to compute the ring of bounded functions in this setting and discuss applications to positive polynomials and the moment problem. Complete results are obtained in special cases, like sets defined by binomial inequalities. We also show that the wild behaviour of certain examples constructed by Krug and by Mondal-Netzer cannot occur in a toric setting.Comment: 19 pages; minor updates and correction