10 research outputs found
Discrete logarithms in curves over finite fields
A survey on algorithms for computing discrete logarithms in Jacobians of
curves over finite fields
Counting classes of special polynomials
Most integers are composite and most univariate polynomials over a finite field are reducible. The Prime Number Theorem and a classical result of GauĂź count the remaining ones, approximately and exactly. In two or more variables, the situation changes dramatically. Most multivariate polynomials are irreducible. We present counting results for some special classes of multivariate polynomials over a finite field, namely the reducible ones, the s-powerful ones (divisible by the s-th power of a nonconstant polynomial), and the relatively irreducible ones (irreducible but reducible over an extension field). These numbers come as exact formulas and as approximations with relative errors that essentially decrease exponentially in the input size. Furthermore, a univariate polynomial f over a field F is decomposable if f = g o h with nonlinear polynomials g and h. It is intuitively clear that the decomposable polynomials form a small minority among all polynomials. The tame case, where the characteristic p of F does not divide n = deg f, is fairly well understood, and the upper and lower bounds on the number of decomposable polynomials of degree n match asymptotically. In the wild case, where p does divide n, the bounds are less satisfactory, in particular when p is the smallest prime divisor of n and divides n exactly twice. There is an obvious inclusion-exclusion formula for counting. The main issue is then to determine, under a suitable normalization, the number of collisions, where essentially different components (g, h) yield the same f. In the tame case, Ritt's Second Theorem classifies all collisions of two such pairs. We provide a normal form for collisions of any number of compositions with any number of components. This generalization yields an exact formula for the number of decomposable polynomials of degree n coprime to p. For the wild case, we classify all collisions at degree n = p^2 and obtain the exact number of decomposable polynomials of degree p^2
On exotic and perverse-coherent sheaves
Exotic sheaves are certain complexes of coherent sheaves on the cotangent bundle of the flag variety of a reductive group. They are closely related to perverse-coherent sheaves on the nilpotent cone. This expository article includes the definitions of these two categories, applications, and some structure theory, as well as detailed calculations for SL .
On Algebraic Singularities, Finite Graphs and D-Brane Gauge Theories: A String Theoretic Perspective
In this writing we shall address certain beautiful inter-relations between
the construction of 4-dimensional supersymmetric gauge theories and resolution
of algebraic singularities, from the perspective of String Theory. We review in
some detail the requisite background in both the mathematics, such as
orbifolds, symplectic quotients and quiver representations, as well as the
physics, such as gauged linear sigma models, geometrical engineering,
Hanany-Witten setups and D-brane probes.
We investigate aspects of world-volume gauge dynamics using D-brane
resolutions of various Calabi-Yau singularities, notably Gorenstein quotients
and toric singularities. Attention will be paid to the general methodology of
constructing gauge theories for these singular backgrounds, with and without
the presence of the NS-NS B-field, as well as the T-duals to brane setups and
branes wrapping cycles in the mirror geometry. Applications of such diverse and
elegant mathematics as crepant resolution of algebraic singularities,
representation of finite groups and finite graphs, modular invariants of affine
Lie algebras, etc. will naturally arise. Various viewpoints and generalisations
of McKay's Correspondence will also be considered.
The present work is a transcription of excerpts from the first three volumes
of the author's PhD thesis which was written under the direction of Prof. A.
Hanany - to whom he is much indebted - at the Centre for Theoretical Physics of
MIT, and which, at the suggestion of friends, he posts to the ArXiv pro hac
vice; it is his sincerest wish that the ensuing pages might be of some small
use to the beginning student.Comment: 513 pages, 71 figs, Edited Excerpts from the first 3 volumes of the
author's PhD Thesi