1,356 research outputs found
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
Representation Growth and Rational Singularities of the Moduli Space of Local Systems
We relate the asymptotic representation theory of and
the singularities of the moduli space of -local systems on a smooth
projective curve, proving new theorems about both. Regarding the former, we
prove that, for every d, the number of n-dimensional representations of
grows slower than , confirming a conjecture of
Larsen and Lubotzky. Regarding the latter, we prove that the moduli space of
-local systems on a smooth projective curve of genus at least 12 has
rational singularities. Most of our results apply more generally to semi-simple
algebraic groups.
For the proof, we study the analytic properties of push forwards of smooth
measures under algebraic maps. More precisely, we show that such push forwards
have continuous density if the algebraic map is flat and all of its fibers have
rational singularities.Comment: preliminary version, comments are welcome. v2. Revised version, now
covering all semi simple group
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