Showing papers on "Generic polynomial published in 2011"

Computational aspects of modular forms and Galois representations : how one can compute in polynomial time the value of Ramanujan's tau at a prime

Abstract: Modular forms are tremendously important in various areas of mathematics, from number theory and algebraic geometry to combinatorics and lattices. Their Fourier coefficients, with Ramanujan's tau-function as a typical example, have deep arithmetic significance. Prior to this book, the fastest known algorithms for computing these Fourier coefficients took exponential time, except in some special cases. The case of elliptic curves (Schoof's algorithm) was at the birth of elliptic curve cryptography around 1985. This book gives an algorithm for computing coefficients of modular forms of level one in polynomial time. For example, Ramanujan's tau of a prime number p can be computed in time bounded by a fixed power of the logarithm of p. Such fast computation of Fourier coefficients is itself based on the main result of the book: the computation, in polynomial time, of Galois representations over finite fields attached to modular forms by the Langlands program. Because these Galois representations typically have a nonsolvable image, this result is a major step forward from explicit class field theory, and it could be described as the start of the explicit Langlands program. The computation of the Galois representations uses their realization, following Shimura and Deligne, in the torsion subgroup of Jacobian varieties of modular curves. The main challenge is then to perform the necessary computations in time polynomial in the dimension of these highly nonlinear algebraic varieties. Exact computations involving systems of polynomial equations in many variables take exponential time. This is avoided by numerical approximations with a precision that suffices to derive exact results from them. Bounds for the required precision--in other words, bounds for the height of the rational numbers that describe the Galois representation to be computed--are obtained from Arakelov theory. Two types of approximations are treated: one using complex uniformization and another one using geometry over finite fields. The book begins with a concise and concrete introduction that makes its accessible to readers without an extensive background in arithmetic geometry. And the book includes a chapter that describes actual computations.

Computational Aspects of Modular Forms and Galois Representations

Abstract: Modular forms are tremendously important in various areas of mathematics, from number theory and algebraic geometry to combinatorics and lattices. Their Fourier coefficients, with Ramanujan's tau-function as a typical example, have deep arithmetic significance. Prior to this book, the fastest known algorithms for computing these Fourier coefficients took exponential time, except in some special cases. The case of elliptic curves (Schoof's algorithm) was at the birth of elliptic curve cryptography around 1985. This book gives an algorithm for computing coefficients of modular forms of level one in polynomial time. For example, Ramanujan's tau of a prime number p can be computed in time bounded by a fixed power of the logarithm of p. Such fast computation of Fourier coefficients is itself based on the main result of the book: the computation, in polynomial time, of Galois representations over finite fields attached to modular forms by the Langlands program. Because these Galois representations typically have a nonsolvable image, this result is a major step forward from explicit class field theory, and it could be described as the start of the explicit Langlands program. The computation of the Galois representations uses their realization, following Shimura and Deligne, in the torsion subgroup of Jacobian varieties of modular curves. The main challenge is then to perform the necessary computations in time polynomial in the dimension of these highly nonlinear algebraic varieties. Exact computations involving systems of polynomial equations in many variables take exponential time. This is avoided by numerical approximations with a precision that suffices to derive exact results from them. Bounds for the required precision--in other words, bounds for the height of the rational numbers that describe the Galois representation to be computed--are obtained from Arakelov theory. Two types of approximations are treated: one using complex uniformization and another one using geometry over finite fields. The book begins with a concise and concrete introduction that makes its accessible to readers without an extensive background in arithmetic geometry. And the book includes a chapter that describes actual computations.

Arboreal Galois representations and uniformization of polynomial dynamics

Abstract: Given a polynomial f of degree d defined over a complete local field, we construct a biholomorphic change of variables defined in a neighbourhood of infinity which transforms the action z->f(z) to the multiplicative action z->z^d. The relation between this construction and the Bottcher coordinate in complex polynomial dynamics is similar to the relation between the complex uniformization of elliptic curves, and Tate's p-adic uniformization. Specifically, this biholomorphism is Galois equivariant, reducing certain questions about the Galois theory of preimages by f to questions about multiplicative Kummer theory.

Discriminant of system of equations

Abstract: What polynomial in the coefficients of a system of algebraic equations should be called its discriminant? We prove a package of facts that provide a possible answer Let us call a system typical, if the homeomorphic type of its set of solutions does not change as we perturb its (non-zero) coefficients The set of all atypical systems turns out to be a hypersurface in the space of all systems of k equations in n variables, whose monomials are contained in k given finite sets This hypersurface B contains all systems that have a singular solution, this stratum is conventionally called the discriminant, and the codimension of its components has not been fully understood yet (eg dual defect toric varieties are not classified), so the purity of dimension of B looks somewhat surprising We deduce it from a similar tropical purity fact A generic system of equations in a component B_i of the hypersurface B differs from a typical system by the Euler characteristic of its set of solutions Regarding the difference of these Euler characteristics as the multiplicity of B_i, we turn B into an effective divisor, whose equation we call the Euler discriminant by the following reasons Firstly, it vanishes exactly at those systems that have a singular solution (possibly at infinity) Secondly, despite its topological definition, there is a simple linear-algebraic formula for it, and a positive formula for its Newton polytope Thirdly, it interpolates many classical objects (sparse resultant, A-determinant, discriminant of deformation) and inherits many of their nice properties This allows to specialize our results to generic polynomial maps: the bifurcation set of a dominant polynomial map, whose components are generic linear combinations of finitely many monomials, is always a hypersurface, and a generic atypical fiber of such a map differs from a typical one by its Euler characteristic

Hopf Galois structures on Kummer extensions of prime power degree

Abstract: Let K be a field of characteristic not p (an odd prime), containing a primitive p n -th root of unity , and let L = K(z) with x p n a the minimal polynomial of z over K: thus L|K is a Kummer extension, with cyclic Galois group G = h i acting on L via (z) = z . T. Kohl, 1998, showed that L|K has p n 1 Hopf Galois structures. In this paper we describe these Hopf Galois structures.

Nonsolvable polynomials with field discriminant 5 a

Abstract: We present the first explicitly known polynomials in Z(x) with nonsolvable Galois group and field discriminant of the form ±p A for p 7 a prime. Our main polyno- mial has degree 25, Galois group of the form PSL2(5) 5 .10, and field discriminant 5 69 . A closely related polynomial has degree 120, Galois group of the form SL2(5) 5 .20, and field discriminant 5 311 . We completely describe 5-adic behavior, finding in particular that the root discriminant of both splitting fields is 125·5 1/12500 124.984 and the class number of the latter field is divisible by 5 4 .

A Threshold for a Polynomial Solution of n2SAT

Abstract: The nSAT problem is a classical nP-complete problem even for monotone, Horn and two conjunctive formulas (the last known as n2SAT). We present a novel branch and bound algorithm to solve the n2SAT problem exactly. Our procedure establishes a new threshold where n2SAT can be computed in polynomial time. We show that for any 2-CF formula F with n variables where n2SAT(F) ≤ p(n), for some polynomial p, n2SAT(F) is computed in polynomial time. This is a new way to measure the degree of difficulty for solving n2SAT and, according to such measure our algorithm allows to determine a boundary between ‘hard’ and ‘easy’ instances of the n2SAT problem.

On the Computation of Differential Resultants

Abstract: The definition of the differential resultant of a set of ordinary differential polynomials is reviewed and its computation via determinants is revisited, using a modern language. This computation is also extended to differential homogeneous resultants of homogeneous ordinary differential polynomials. A numeric example is included and an example of the application of elimination theory to biological modelling is revisited, in terms of differential resultants.

An explicit candidate for the set of Steinitz classes of tame Galois extensions with fixed Galois group of odd order

Abstract: Given a finite group G and a number field k, a well-known conjecture asserts that the set R_t(k,G) of Steinitz classes of tame G-Galois extensions of k is a subgroup of the ideal class group of k. In this paper we investigate an explicit candidate for R_t(k,G), when G is of odd order. More precisely, we define a subgroup W(k,G) of the class group of k and we prove that R_t(k,G) is contained in W(k,G). We show that equality holds for all groups of odd order for which a description of R_t(k,G) is known so far. Furthermore, by refining techniques introduced in arXiv:0910.5080v1, we use the Shafarevich-Weil Theorem in cohomological class field theory, to construct some tame Galois extensions with given Steinitz class. In particular, this allows us to prove the equality R_t(k,G)=W(k,G) when G is a group of order dividing l^4, where l is an odd prime.

Armstrong systems and Galois connections

Abstract: In the paper [1], it is proved that any Galois connection (f, g) on a complete lattice made an Armstrong system F (f, g) . We prove in this short note that the converse holds, that is, for a given Armstrong system R, we can make a Galois connection (φ R , ψ R ) and the original Armstrong system R is identical with the induced Armstrong system F (φR, ψR) by the Galois connection (φ R , ψ R ). This means that Armstrong systems and Galois connections show us two faces of one thing.

Polynomial endomorphisms over finite fields: experimental results

Abstract: Given a finite field Fq and n 2 N � , one could try to compute all polynomial endomorphisms F n −! F n up to a certain degree with a specific property. We consider the case n = 3. If the degree is low (like 2,3, or 4) and the finite field is small (q � 7) then some of the computations are still feasible. In this article we study the following properties of endomorphisms: being a bijection of F n −! F n , being a polynomial automorphism, being a Mock automorphism, and being a locally finite polynomial automorphism. In the resulting tables, we point out a few interesting objects, and pose some interesting conjectures which surfaced through our computations.

A Structure of Galois Extensions with an Inner Galois Group

Abstract: Let B be a Galois extension of B G with an inner Galois group G where G = {gi |gi(x )= UixU −1 i for some Ui ∈ B and for all x ∈ B, i =1 ,2, ··· ,nfor some integer n invertible in B}. Then B is a composition of two Galois extensions B ⊃ B K with an inner Abelian Galois group K and B K ⊃ B G with an inner Galois group G/K where K = {g ∈ G |g(Ui )= Ui for each i}. Descriptions of B ⊃ B K and B K ⊃ B G are given.

Thoughts on the reduced Whitehead group of the Iwasawa algebra

Abstract: Let l be an odd prime and K/k a Galois extension of totally real number fields with Galois group G such that K/k_\infty and k/Q are finite. We reduce the conjectured triviality of the reduced Whitehead group SK_1(QG) of the algebra QG=Quot(\Lambda G) with the Iwasawa algebra \Lambda G = Z_l[[G]] to the case of pro-l Galois groups G and finite unramified coefficient extensions.

Polynomial Invariants of Certain Pseudo-Symplectic Groups Over Finite Fields of Characteristic Two

Abstract: Let F q be a finite field of characteristic two, S be a nonsingular non-alternate symmetric matrix over F q and Ps n (F q , S) be the associated pseudo-symplectic group. Let Ps n (F q , S) act linearly on the polynomial ring F q [x 1,…, x n ]. In this note, we find an explicit set of generators of the ring of invariants of Ps n (F q , S) for n = 2, 4 and 2ν +1. In particular, the results assert that the ring of invariants of Ps 4(F q , S) is not a polynomial algebra but is an example of hypersurface and the ring of invariants of Ps 2ν+1(F q , S) is a complete intersection.

Field Theory and Galois Theory

Abstract: The general solutions of linear and quadratic polynomial in one variable were known centuries before. For cubic and quartic equations also the general solutions are provided by Cardano's and Ferrari's methods respectively. In 19th century a great work has been done to find general solution of a general polynomial by radicals. However there was no success even after efforts of many great mathematicians of that time. Eventually work by Able and Galois gives satisfactory solution and complete understanding of this problem.Galois Theory provides a connection between Field theory and Group theory, which in turn useful to convert problems in field theory into Group theory, which are better understood and easy to handle. Galois theory not only provide answer to the problem discussed above but also explains why the general solution exists for polynomials with degree less then or equal to 4. In his original work, Galois used permutation groups to describe relations between roots of the polynomial. In modern approach, developed by Artin, Dedekind etc., involves study of automorphisms of field extensions.

Perturbation analysis of eigenvalues of polynomial matrices smoothly depending on parameters

Abstract: Let P(λ) = Σi=0kλiAi(p) be a family of monic polynomial matrices smoothly dependent on a vector of real parameters p = (p1,..., pn). In this work we study behavior of a multiple eigenvalue of the monic polynomial family P(λ).