43,542 research outputs found
Recognizing Graph Theoretic Properties with Polynomial Ideals
Many hard combinatorial problems can be modeled by a system of polynomial
equations. N. Alon coined the term polynomial method to describe the use of
nonlinear polynomials when solving combinatorial problems. We continue the
exploration of the polynomial method and show how the algorithmic theory of
polynomial ideals can be used to detect k-colorability, unique Hamiltonicity,
and automorphism rigidity of graphs. Our techniques are diverse and involve
Nullstellensatz certificates, linear algebra over finite fields, Groebner
bases, toric algebra, convex programming, and real algebraic geometry.Comment: 20 pages, 3 figure
Universality for Random Tensors
We prove two universality results for random tensors of arbitrary rank D. We
first prove that a random tensor whose entries are N^D independent, identically
distributed, complex random variables converges in distribution in the large N
limit to the same limit as the distributional limit of a Gaussian tensor model.
This generalizes the universality of random matrices to random tensors.
We then prove a second, stronger, universality result. Under the weaker
assumption that the joint probability distribution of tensor entries is
invariant, assuming that the cumulants of this invariant distribution are
uniformly bounded, we prove that in the large N limit the tensor again
converges in distribution to the distributional limit of a Gaussian tensor
model. We emphasize that the covariance of the large N Gaussian is not
universal, but depends strongly on the details of the joint distribution.Comment: Final versio
Counting generalized Jenkins-Strebel differentials
We study the combinatorial geometry of "lattice" Jenkins--Strebel
differentials with simple zeroes and simple poles on and of the
corresponding counting functions. Developing the results of M. Kontsevich we
evaluate the leading term of the symmetric polynomial counting the number of
such "lattice" Jenkins-Strebel differentials having all zeroes on a single
singular layer. This allows us to express the number of general "lattice"
Jenkins-Strebel differentials as an appropriate weighted sum over decorated
trees.
The problem of counting Jenkins-Strebel differentials is equivalent to the
problem of counting pillowcase covers, which serve as integer points in
appropriate local coordinates on strata of moduli spaces of meromorphic
quadratic differentials. This allows us to relate our counting problem to
calculations of volumes of these strata . A very explicit expression for the
volume of any stratum of meromorphic quadratic differentials recently obtained
by the authors leads to an interesting combinatorial identity for our sums over
trees.Comment: to appear in Geometriae Dedicata. arXiv admin note: text overlap with
arXiv:1212.166
The large sieve, monodromy and zeta functions of curves
We prove a large sieve statement for the average distribution of Frobenius
conjugacy classes in arithmetic monodromy groups over finite fields. As a first
application we prove a stronger version of a result of Chavdarov on the
``generic'' irreducibility of the numerator of the zeta functions in a family
of curves with large monodromy.Comment: 30 page
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