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.Comment: 1 figur

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

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

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.