2,543 research outputs found
Lehmer points and visible points on affine varieties over finite fields
Let be an absolutely irreducible affine variety over . A
Lehmer point on is a point whose coordinates satisfy some prescribed
congruence conditions, and a visible point is one whose coordinates are
relatively prime. Asymptotic results for the number of Lehmer points and
visible points on are obtained, and the distribution of visible points into
different congruence classes is investigated.Comment: 16 pages, accepted for publication in Math. Proc. Cambridge Philos.
Soc. Theorem 2 of this paper is improved from the published versio
Generic properties of subgroups of free groups and finite presentations
Asymptotic properties of finitely generated subgroups of free groups, and of
finite group presentations, can be considered in several fashions, depending on
the way these objects are represented and on the distribution assumed on these
representations: here we assume that they are represented by tuples of reduced
words (generators of a subgroup) or of cyclically reduced words (relators).
Classical models consider fixed size tuples of words (e.g. the few-generator
model) or exponential size tuples (e.g. Gromov's density model), and they
usually consider that equal length words are equally likely. We generalize both
the few-generator and the density models with probabilistic schemes that also
allow variability in the size of tuples and non-uniform distributions on words
of a given length.Our first results rely on a relatively mild prefix-heaviness
hypothesis on the distributions, which states essentially that the probability
of a word decreases exponentially fast as its length grows. Under this
hypothesis, we generalize several classical results: exponentially generically
a randomly chosen tuple is a basis of the subgroup it generates, this subgroup
is malnormal and the tuple satisfies a small cancellation property, even for
exponential size tuples. In the special case of the uniform distribution on
words of a given length, we give a phase transition theorem for the central
tree property, a combinatorial property closely linked to the fact that a tuple
freely generates a subgroup. We then further refine our results when the
distribution is specified by a Markovian scheme, and in particular we give a
phase transition theorem which generalizes the classical results on the
densities up to which a tuple of cyclically reduced words chosen uniformly at
random exponentially generically satisfies a small cancellation property, and
beyond which it presents a trivial group
Symmetries of statistics on lattice paths between two boundaries
We prove that on the set of lattice paths with steps N=(0,1) and E=(1,0) that
lie between two fixed boundaries T and B (which are themselves lattice paths),
the statistics `number of E steps shared with B' and `number of E steps shared
with T' have a symmetric joint distribution. To do so, we give an involution
that switches these statistics, preserves additional parameters, and
generalizes to paths that contain steps S=(0,-1) at prescribed x-coordinates.
We also show that a similar equidistribution result for path statistics follows
from the fact that the Tutte polynomial of a matroid is independent of the
order of its ground set. We extend the two theorems to k-tuples of paths
between two boundaries, and we give some applications to Dyck paths,
generalizing a result of Deutsch, to watermelon configurations, to
pattern-avoiding permutations, and to the generalized Tamari lattice. Finally,
we prove a conjecture of Nicol\'as about the distribution of degrees of k
consecutive vertices in k-triangulations of a convex n-gon. To achieve this
goal, we provide a new statistic-preserving bijection between certain k-tuples
of non-crossing paths and k-flagged semistandard Young tableaux, which is based
on local moves reminiscent of jeu de taquin.Comment: Small typos corrected, and journal reference and grant info adde
Polynomials with prescribed bad primes
We tabulate polynomials in Z[t] with a given factorization partition, bad
reduction entirely within a given set of primes, and satisfying auxiliary
conditions associated to 0, 1, and infinity. We explain how these sets of
polynomials are of particular interest because of their role in the
construction of nonsolvable number fields of arbitrarily large degree and
bounded ramification. Finally we discuss the similar but technically more
complicated tabulation problem corresponding to removing the auxiliary
conditions.Comment: 26 pages, 3 figure
Statistics for traces of cyclic trigonal curves over finite fields
We study the variation of the trace of the Frobenius endomorphism associated
to a cyclic trigonal curve of genus g over a field of q elements as the curve
varies in an irreducible component of the moduli space. We show that for q
fixed and g increasing, the limiting distribution of the trace of the Frobenius
equals the sum of q+1 independent random variables taking the value 0 with
probability 2/(q+2) and 1, e^{(2pi i)/3}, e^{(4pi i)/3} each with probability
q/(3(q+2)). This extends the work of Kurlberg and Rudnick who considered the
same limit for hyperelliptic curves. We also show that when both g and q go to
infinity, the normalized trace has a standard complex Gaussian distribution and
how to generalize these results to p-fold covers of the projective line.Comment: 30 pages, added statement and sketch of proof in Section 7 for
generalization of results to p-fold covers of the projective line, the final
version of this article will be published in International Mathematics
Research Notice
Integration with respect to the Haar measure on unitary, orthogonal and symplectic group
We revisit the work of the first named author and using simpler algebraic
arguments we calculate integrals of polynomial functions with respect to the
Haar measure on the unitary group U(d). The previous result provided exact
formulas only for 2d bigger than the degree of the integrated polynomial and we
show that these formulas remain valid for all values of d. Also, we consider
the integrals of polynomial functions on the orthogonal group O(d) and the
symplectic group Sp(d). We obtain an exact character expansion and the
asymptotic behavior for large d. Thus we can show the asymptotic freeness of
Haar-distributed orthogonal and symplectic random matrices, as well as the
convergence of integrals of the Itzykson-Zuber type
Fluctuations for analytic test functions in the Single Ring Theorem
We consider a non-Hermitian random matrix whose distribution is invariant
under the left and right actions of the unitary group. The so-called Single
Ring Theorem, proved by Guionnet, Krishnapur and Zeitouni, states that the
empirical eigenvalue distribution of converges to a limit measure supported
by a ring . In this text, we establish the convergence in distribution of
random variables of the type where is analytic on and the
Frobenius norm of has order . As corollaries, we obtain central
limit theorems for linear spectral statistics of (for analytic test
functions) and for finite rank projections of (like matrix entries). As
an application, we locate outliers in multiplicative perturbations of .Comment: 29 pages, 1 figure. In Version v2, we slightly modified the
assumptions, in order to fix a problem un the control of the tails (see
Assumption 2.3). In v3, some minors typos were corrected. In v4, some
explanations were added in the introduction and some typos were corrected. To
appear in Indiana Univ. Math.
Biased statistics for traces of cyclic p-fold covers over finite fields
In this paper, we discuss in more detail some of the results on the statistics of the trace of the Frobenius endomorphism associated to cyclic p-fold covers of the projective line that were presented in [1]. We also show new findings regarding statistics associated to such curves where we fix the number of zeros in some of the factors of the equation in the affine model
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