73 research outputs found
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
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
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