Showing papers on "Generic polynomial published in 2008"

Functoriality and the inverse Galois problem

Abstract: We prove that, for any primeand any even integer n, there are infinitely many exponents k for which PSp n (Fk) appears as a Galois group over Q. This generalizes a result of Wiese from 2006, which inspired this paper.

Polynomial phase masks for extending the depth of field of a microscope.

Abstract: A polynomial phase mask is designed and fabricated for enhancing the depth of field of a microscope by more than tenfold. A generic polynomial of degree 31 that consists of 16 odd power terms is optimized by simulated annealing with a realistic average modulation transfer function (MTF) iteratively set as the target MTF. Optical experimental results are shown.

Closed galois field combination

Abstract: A method is provided for combining two or more input sequences in a communications system to increase a repetition period of the input sequences in a resource-efficient manner. The method includes a receiving step, a mapping step, and a generating step. The receiving step involves receiving a first number sequence and a second number sequence, each expressed in a Galois field GF¬p k|. The mapping step involves mapping the first and second number sequences to a Galois extension field GF¬p k+1|. The generating step involves generating an output sequence by combining the first number sequence with the second number sequence utilizing a Galois field multiplication operation in the Galois extension field GF¬p k+1|. p is a prime number. k is an integer. p k+1 defines a finite field size of the Galois extension field GF¬p k+1|.

POLYNOMIALS WITH ROOTS IN Qp FOR ALL p

Abstract: Let f(x) be a monic polynomial in Z[x] with no rational roots but with roots in Qp for all p, or equivalently, with roots mod n for all n. It is known that f(x) cannot be irreducible but can be a product of two or more irreducible polynomials, and that if f(x) is a product of m > 1 irreducible polynomials, then its Galois group must be a union of conjugates of m proper subgroups. We prove that for any m > 1, every finite solvable group that is a union of conjugates of m proper subgroups (where all these conjugates have trivial intersection) occurs as the Galois group of such a polynomial, and that the same result (with m = 2) holds for all Frobenius groups. It is also observed that every nonsolvable Frobenius group is realizable as the Galois group of a geometric, i.e. regular, extension of Q(t).

Cryptographic processing and processors

Abstract: A method of performing a cryptographic process on data, the cryptographic process treating a quantity of the data as an element of a Galois field GF(λ k ), where k=rs, the method comprising: isomorphically mapping the element of the Galois field GF(λ k ) to an s-tuple of elements of a Galois field GF(λ′); and representing and processing each of the elements of the s-tuple of elements of the Galois field GF(λ′) in the form of one or more respective n-of-m codewords, where an n-of-m codeword comprises n 1-bits and m-n 0-bits, where m and n are predetermined positive integers and n is less than m.

Large Selmer groups over number fields

Abstract: Let p be a prime number and M a quadratic number field, M not equal to Q(\sqrt{p}) if p is congruent to 1 modulo 4. We will prove that for any positive integer d there exists a Galois extension F/Q with Galois group D_{2p} and an elliptic curve E/Q such that F contains M and the p-Selmer group of E/F has size at least p^d.

Multi-modular algorithm for computing the splitting field of a polynomial

Abstract: Let f be a univariate monic integral polynomial of degree n and let (α1, ..., αn) be an n-tuple of its roots in an algebraic closure Q of Q. Obtaining an algebraic representation of the splitting field Q(α1, ..., αn) of f is a question of first importance in effective Galois theory. For instance, it allows us to manipulate symbolically the roots of f. In this paper, we propose a new method based on multi-modular strategy. Actually, we provide algorithms for this task which return a triangular set encoding the splitting ideal of f. We examine the ability/practicality of the method by experiments on a real computer and study its complexity.

Relative brauer groups in characteristic

Abstract: This paper gives a description of the relative Brauer group Br(E/F) when F has characteristic p, [E : F] = p, and the Galois group Gal(E 1 /F) is solvable, where E 1 is the Galois closure of E over F.

GF(2 m ) multiplier using Polynomial Residue Number System

Abstract: This paper studies the polynomial residue representation of Galois field (2m) elements and polynomial residue arithmetic (PRA), according to which a novel approach of performing GF(2m) multiplication using polynomial residue number system (PRNS) is introduced. A channel-serial and a channel-parallel architecture of the PRNS multiplier over GF(2m) are presented. Conclusion is drawn by comparing the synthesis results of these two architectures.

Dessins d'enfants and differential equations

Abstract: We state and solve a discrete version of the classical Riemann-Hilbert problem. In particular, we associate a Riemann-Hilbert problem to every dessin d'enfants. We show how to compute the solution for a dessin that is a tree. This amounts to finding a Fuchsian differential equation satisfied by the local inverses of a Shabat polynomial. We produce a uni- versal annihilating operator for the inverses of a generic polynomial. We classify those plane trees that have a representation by Mobius transformations and those that have a linear rep- resentation of dimension at most two. This yields an analogue for trees of Schwarz's classical list, that is, a list of those plane trees whose Riemann-Hilbert problem has a hypergeometric solution of order at most two.

On H-Separable and Galois Polynomials of Degree p in Skew Polynomial Rings

Abstract: Let B be a ring with identity 1, D a derivation of B, and B[X; D] the skew polynomial ring such that αX = Xα+ D(α) for each α ∈ B. Then conditions are given for X p − Xa− b ∈ B[X; D ]t o be anH-separable and Galois polynomial where p is a prime integer.

On the Geometry of p-Typical Covers in Characteristic p

Abstract: For p a prime, a p-typical cover of a connected scheme on which p = 0 is a finite ´ etale cover whose monodromy group (i.e., the Galois group of its normal closure) is a p-group. The geometry of such covers exhibits some unexpectedly pleasant behaviors; building on work of Katz, we demonstrate some of these. These include a criterion for when a morphism induces an isomorphism of the p-typi- cal quotients of thefundamental groups, and a decomposition theorem for p-typical covers of polynomial rings over an algebraically closed field.

An optimal control application in power electronics using numerical algebraic geometry

Abstract: We present an optimal control application in power electronics using the homotopy continuation method for solving systems of polynomial equations The proposed approach breaks the computations associated with the optimal control problem into two parts, an off-line and an on-line In the off-line part, the approach solves a generic polynomial system by means of a linear homotopy and stores its solution Then, the on-line part uses this solution and, given the initial state value, it calculates by means of a coefficient parameter homotopy the optimal control input of the problem The approach exhibits a probability-one guarantee of finding the global optimal solution to the problem at hand

Generic Jordan Polynomials

Abstract: The universal multiplication envelope 𝒰 ℳ ℰ(J) of a Jordan system J (algebra, triple, or pair) encodes information about its linear actions—all of its possible actions by linear transformations on outer modules M (equivalently, on all larger split null extensions J ⊕ M). In this article, we study all possible actions, linear and nonlinear, on larger systems. This is encoded in the universal polynomial envelope 𝒰 𝒫 ℰ(J), which is a system containing J and a set X of indeterminates. Its elements are generic polynomials in X with coefficients in the system J, and it encodes information about all possible multiplications by J on extensions . The universal multiplication envelope is recovered as the “linear part,” the elements homogeneous of degree 1 in some variable x. We are especially interested in generic polynomial identities, free Jordan polynomials p(x 1,…, x n ; y 1,…, y m ) which vanish for particular a j ∈ J and all possible x i in all , i.e., such that the generic polynomial p(x 1,…, x n ; a 1,…, a...

Minimality and ergodicity of a generic analytic foliation of 2

Abstract: It is well known that a generic polynomial foliation of 2 is minimal and ergodic. In this paper we prove an analogous result for analytic foliations

Galois modular representation of associated Jacobians in the tamely ramified cyclic case

Abstract: Let L/K be an l-cyclic extension with Galois group G of algebraic function fields over an algebraically closed field k of characteristic p ≠ l. In this paper, the \({\mathbb{Z}_{\ell}[G]}\)-module structure of the l-torsion of the Jacobian associated to L is explicitly determined.

Fundamental Properties of the Galois Correspondence

Abstract: Problem Statement: Let K is the splitting field of a polynomial f(x) over a field F and αn be the roots of f in K. Let G be embedded as a subgroup of the symmetric group ς. We determined the Galois group G, and the subgroup. Approach: computed some auxiliary polynomials that had roots in K, where the permutation of a set was considered distinct. The Galois Theory was deduced using the primitive element and Splitting theorems. Results: The Galois extension K/L to identity L and its Galois group is a subgroup of G. which was referred to as the main theorem which we proved. Conclusion: Hence the findings suggest the need for computing more auxiliary polynomials that have roots.

Integer Points in Backward Orbits

Abstract: A theorem of J. Silverman states that a forward orbit of a rational map $\phi(z)$ on $\mathbb P^1(K)$ contains finitely many $S$-integers in the number field $K$ when $(\phi\circ\phi)(z)$ is not a polynomial. We state an analogous conjecture for the backward orbits using a general $S$-integrality notion based on the Galois conjugates of points. This conjecture is proven for the map $\phi(z)=z^d$, and consequently Chebyshev polynomials, by uniformly bounding the number of Galois orbits for $z^n-\beta$ when $\beta ot =0$ is a non-root of unity. In general, our conjecture is true provided that the number of Galois orbits for $\phi^n(z)-\beta$ is bounded independently of $n$.

The Galois group of the Eisenstein polynomial X 5 + aX + a

Abstract: Let p be a rational prime and let a be an integer which is divisible by p exactly to the first power. Then the Galois group \(G = \hbox{Gal}_{{\mathbb{Q}}}(f)\) of the Eisenstein polynomial f = Xp + aX + a is known to be either the full symmetric group Sp or the affine group A(1, p), and it is conjectured that always G = Sp. In this note we settle this conjecture for p = 5 and, answering a question by J.-P. Serre, we show that this does not carry over when replacing the integer a by some rational number with 5-adic valuation equal to 1.

Aes algorithm processing method and processors resistant to differential power analysis attack

Abstract: A method of performing a cryptographic process on data, the cryptographic process treating a quantity of the data as an element of a Galois field GF(λk), where k=rs, the method comprising: isomorphically mapping the element of the Galois field GF(λk) to an s-tuple of elements of a Galois field GF(λ'); and representing and processing each of the elements of the s-tuple of elements of the Galois field GF(λ') in the form of one or more respective n-of-m codewords, where an n-of-m codeword comprises n 1-bits and m - n 0-bits, where m and n are predetermined positive integers and n is less than m.

Note sur les corps 2-rationnels

Abstract: We compute the Galois group of the maximal 2-ramified pro-2-extension of a 2-rational number field

Quantum systems with positions and momenta on a Galois field

Abstract: Quantum systems with positions and momenta in the Galois field GF(pe), are considered. The Heisenberg-Weyl group of displacements and the Sp(2,GF(pe)) group of symplectic transformations, are studied. Frobenius symmetries, are a unique feature of these systems and lead to constants of motion. The engineering of such systems from l spins with j = (p - 1)/2, which are coupled in a particular way, is discussed.

On the Dec group of finite abelian Galois extensions over global fields

Abstract: If K/F is a finite abelian Galois extension of global fields whose Galois group has exponent t, we prove that there exists a short exact sequence that has as a consequence that if t is square free, then Dec(K/F)=Br_{t}(K/F) which we use to show that prime exponent division algebras over Henselian valued fields with global residue fields are isomorphic to a tensor product of cyclic algebras. Finally, we construct a counterexample to the result for higher exponent division algebras.

Galois group of generalized inverse polynomial modules

Abstract: Given an injective envelope E of a left R-module M, there is an associative Galois group Gal(ϕ) Let R be a left noetherian ring and E be an injective envelope of M, then there is an injective envelope E(x−1) of an inverse polynomial module M(x−1) as a left R(x)-module and we can define an associative Galois group Gal(ϕ(x−1)) In this paper we extend the Galois group of inverse polynomial module and can get Gal(ϕ(x−s)), where S is a submonoid of N (the set of all natural numbers)