162 research outputs found
The Łojasiewicz exponent over a field of arbitrary characteristic
Let K be an algebraically closed field and let K((XQ)) denote the field
of generalized series with coefficients in K. We propose definitions of the local
Łojasiewicz exponent of F = ( f1, . . . , fm) ∈ K[[X, Y ]]m as well as of the
Łojasiewicz exponent at infinity of F = ( f1, . . . , fm) ∈ K[X, Y ]m, which generalize
the familiar case of K = C and F ∈ C{X, Y }m (resp. F ∈ C[X, Y ]m), see
Cha˛dzy´nski and Krasi´nski (In: Singularities, 1988; In: Singularities, 1988; Ann Polon
Math 67(3):297–301, 1997; Ann Polon Math 67(2):191–197, 1997), and prove some
basic properties of such numbers. Namely, we show that in both cases the exponent
is attained on a parametrization of a component of F (Theorems 6 and 7), thus being
a rational number. To this end, we define the notion of the Łojasiewicz pseudoexponent
of F ∈ (K((XQ))[Y ])m for which we give a description of all the generalized
series that extract the pseudoexponent, in terms of their jets. In particular, we show
that there exist only finitely many jets of generalized series giving the pseudoexponent
of F (Theorem 5). The main tool in the proofs is the algebraic version of Newton’s
Polygon Method. The results are illustrated with some explicit examples
On -Simple -Path
An -simple -path is a {path} in the graph of length that passes
through each vertex at most times. The -SIMPLE -PATH problem, given a
graph as input, asks whether there exists an -simple -path in . We
first show that this problem is NP-Complete. We then show that there is a graph
that contains an -simple -path and no simple path of length greater
than . So this, in a sense, motivates this problem especially
when one's goal is to find a short path that visits many vertices in the graph
while bounding the number of visits at each vertex.
We then give a randomized algorithm that runs in time that solves the -SIMPLE -PATH on a graph with
vertices with one-sided error. We also show that a randomized algorithm
with running time with gives a
randomized algorithm with running time \poly(n)\cdot 2^{cn} for the
Hamiltonian path problem in a directed graph - an outstanding open problem. So
in a sense our algorithm is optimal up to an factor
Tannakian duality for Anderson-Drinfeld motives and algebraic independence of Carlitz logarithms
We develop a theory of Tannakian Galois groups for t-motives and relate this
to the theory of Frobenius semilinear difference equations. We show that the
transcendence degree of the period matrix associated to a given t-motive is
equal to the dimension of its Galois group. Using this result we prove that
Carlitz logarithms of algebraic functions that are linearly independent over
the rational function field are algebraically independent.Comment: 39 page
On the Colmez conjecture for non-abelian CM fields
The Colmez conjecture relates the Faltings height of an abelian variety with complex multiplication by the ring of integers of a CM field E to logarithmic derivatives of Artin L-functions at s=0. In this paper, we prove that if F is any fixed totally real number field of degree [F:ℚ]≥3, then there are infinitely many effective, “positive density” sets of CM extensions E / F such that E/ℚ is non-abelian and the Colmez conjecture is true for E. Moreover, these CM extensions are explicitly constructed to be ramified at arbitrary prescribed sets of prime ideals of F. We also prove that the Colmez conjecture is true for a generic class of non-abelian CM fields called Weyl CM fields, and use this to develop an arithmetic statistics approach to the Colmez conjecture based on counting CM fields of fixed degree and bounded discriminant. We illustrate these results by evaluating the Faltings height of the Jacobian of a genus 2 hyperelliptic curve with complex multiplication by a non-abelian quartic CM field in terms of the Barnes double Gamma function at algebraic arguments. This can be viewed as an explicit non-abelian Chowla–Selberg formula. Our results rely crucially on an averaged version of the Colmez conjecture which was recently proved independently by Andreatta–Goren–Howard–Madapusi Pera and Yuan–Zhang.NFS with grants DMS-1162535 and DMS-1460766UCR::Vicerrectoría de Investigación::Unidades de Investigación::Ciencias Básicas::Centro de Investigaciones en Matemáticas Puras y Aplicadas (CIMPA
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