79 research outputs found
Local Maxima of Quadratic Boolean Functions
How many strict local maxima can a real quadratic function on
have? Holzman conjectured a maximum of . The aim
of this paper is to prove this conjecture. Our approach is via a generalization
of Sperner's theorem that may be of independent interest.Comment: 6 pages, 1 figur
Algebraic and o-minimal flows beyond the cocompact case
Let be an algebraic variety, and let be a discrete subgroup whose real and complex spans
agree. We describe the topological closure of the image of in , thereby extending a result of Peterzil-Starchenko in the case when
is cocompact.
We also obtain a similar extension when is definable
in an o-minimal structure with no restrictions on , and as an
application prove the following conjecture of Gallinaro: for a closed
semi-algebraic (such as a complex algebraic variety)
and the coordinate-wise exponential
map, we have where are
positive-dimensional compact real tori and are
semi-algebraic.Comment: 8 pages, comments welcome
Log-concavity of matroid h-vectors and mixed Eulerian numbers
For any matroid , we compute the Tutte polynomial using the
mixed intersection numbers of certain tautological classes in the combinatorial
Chow ring arising from Grassmannians. Using mixed Hodge-Riemann
relations, we deduce a strengthening of the log-concavity of the h-vector of a
matroid complex, improving on an old conjecture of Dawson whose proof was
announced recently by Ardila, Denham and Huh.Comment: 29 pages, comments welcome
Geometric and o-minimal Littlewood-Offord problems
The classical Erd\H{o}s-Littlewood-Offord theorem says that for nonzero
vectors , any , and uniformly
random , we have
. In this paper we show that
whenever is definable
with respect to an o-minimal structure (for example, this holds when is any
algebraic hypersurface), under the necessary condition that it does not contain
a line segment. We also obtain an inverse theorem in this setting
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