Showing papers on "Generic polynomial published in 2006"

A modular method for computing the splitting field of a polynomial

Abstract: We provide a modular method for computing the splitting field Kf of an integral polynomial f by suitable use of the byproduct of computation of its Galois group Gf by p-adic Stauduhar’s method. This method uses the knowledge of Gf with its action on the roots of f over a p-adic number field, and it reduces the computation of Kf to solving systems of linear equations modulo some powers of p and Hensel liftings. We provide a careful treatment on reducing computational difficulty. We examine the ability/practicality of the method by experiments on a real computer and study its complexity.

Polynomial recurrences and cyclic resultants

Abstract: Let K be an algebraically closed field of characteristic zero and let f ∈ K[x]. The m-th cyclic resultant of f is r m = Res(f , x m - 1). A generic monic polynomial is determined by its full sequence of cyclic resultants; however, the known techniques proving this result give no effective computational bounds. We prove that a generic monic polynomial of degree d is determined by its first 2 d+1 cyclic resultants and that a generic monic reciprocal polynomial of even degree d is determined by its first 2 3 d/2 of them. In addition, we show that cyclic resultants satisfy a polynomial recurrence of length d+ 1. This result gives evidence supporting the conjecture of Sturmfels and Zworski that d + 1 resultants determine f. In the process, we establish two general results of independent interest: we show that certain Toeplitz determinants are sufficient to determine whether a sequence is linearly recurrent, and we give conditions under which a linearly recurrent sequence satisfies a polynomial recurrence of shorter length.

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 universal 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 representation 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.

A generic analytic foliation in ℂ2 has infinitely many cylindrical leaves

Abstract: It is well known that a generic polynomial vector field of degree higher than 2 on the plane has countably many complex limit cycles that are homologically independent on the leaves. In the paper, a similar assertion is proved for analytic vector fields on the complex plane. The proof essentially uses the results of D.S. Volk and T.S. Firsova.

Computation of the splitting field of a dihedral polynomial

Abstract: Let g be a univariate separable polynomial of degree n with coefficients in a computable field K and let (α1, . . . , αn) be an n-tuple of its roots in an algebraic closure K of K. Obtaining an algebraic representation of the splitting field K(α1, . . . , αn) of g is a question of first importance in effective Galois theory. For instance, it allows us to manipulate symbolically the roots of g. In this paper, we focus on the computation of the splitting field of g when its Galois group is a dihedral group. We provide an algorithm for this task which returns a triangular set encoding the relations ideal of g which has degree 2n since the Galois group of g is dihedral. Our algorithm starts from a factorization of g in K[X]/ and constructs the searched triangular set by performing n2 computations of normal forms modulo an ideal of degree 2n.

Inverting bijective polynomial maps over finite fields

Abstract: We study the problem of inverting a bijective polynomial map F: F q n→ F q nover a finite field F q . Our interest mainly stems from the case where F encodes a permutation given by some cryptographic scheme. Given y(0)∈ F q n, we are able to compute the value x(0)∈ F q nfor which F(x(0)) = y(0)holds in time O(LnO(1)δ4) up to logarithmic terms. Here L is the cost of the evaluation of F and δ is a geometric invariant associated to the graph of the polynomial map F, called its degree.

Computation of the Galois group of a polynomial with rational coefficients. II

Abstract: A new method, which enables us to compute rather efficiently the Galois group of a polynomial over ℚ or over ℤ, is presented. Reductions of this polynomial with respect to different prime modules are studied, and the information obtained is used for the calculation of the Galois group of the initial polynomial. This method uses an original modification of the Chebotarev density theorem, and it is in essence a probabilistic method. The irreducibility of the polynomial under consideration is not assumed. The appendix to this paper contains tables, which enable us to find the Galois group of polynomials of degree less than or equal to 10 as a subgroup of the symmetric group. Here the final part of the paper is published. The first part is contained in a previous issue (see Vol. 134, No. 6 (2006)). Bibliography: 10 titles.

Artin-schreier extensions and their applications

Abstract: A Galois extension E/F of fields is called a cyclic extension if the Galois group is cyclic. Assume that p > 0 is the characteristic of our fields and n is the degree of the field extension E/F. If n is relatively prime to p, and there is a primitive n th root of unity in F, then E/F is a Kummer extension, i.e. E = F(y) with y n ∈ F. If n = p, then E/F is an Artin-Schreier extension, i.e. E = F(y) with y p y ∈ F. Finally, if n = p a for a > 1, then the extension E/F can be described in terms of Witt vectors. For these facts, see [34, Section VI.7].

A polynomial time nilpotence test for galois groups and related results

Abstract: We give a deterministic polynomial-time algorithm to check whether the Galois group Gal(f) of an input polynomial f(X)∈ℚ[X] is nilpotent: the running time is polynomial in size(f). Also, we generalize the Landau-Miller solvability test to an algorithm that tests if Gal(f) is in Γd: this algorithm runs in time polynomial in size(f) and nd and, moreover, if Gal(f)∈Γd it computes all the prime factors of # Gal(f).

Optimization and approximation problems related to polynomial system solving

Abstract: We outline some current work in real number complexity theory with a focus on own results. The topics discussed are all located in the area of polynomial system solving. First, we concentrate on a combinatorial optimization problem related to homotopy methods for solving numerically generic polynomial systems. Then, approximation problems are discussed in relation with Probabilistically Checkable Proofs over the real numbers.

Detecting pro-p-groups that are not absolute Galois groups

Abstract: We present several constraints on the absolute Galois groups G_F of fields F containing a primitive pth root of unity, using restrictions on the cohomology of index p normal subgroups from a previous paper by three of the authors. We first classify all maximal p-elementary abelian-by-order p quotients of such G_F. In the case p>2, each such quotient contains a unique closed index p elementary abelian subgroup. This seems to be the first case in which one can completely classify nontrivial quotients of absolute Galois groups by characteristic subgroups of normal subgroups. We then derive analogues of theorems of Artin-Schreier and Becker for order p elements of certain small quotients of G_F. Finally, we construct a new family of pro-p-groups which are not absolute Galois groups over any field F.

Galois deformation and l-invariant

Abstract: 1. Lecture 2 The notation is as in the first lecture (F: a totally real field, p> 2 is a fixed prime). For simplicity, we assume that p splits completely in F/Q. We start with a Galois representation F : Gal(Q/F) ! GL2(W) associated to a Hilbert modular form (on GL(2)/F) with coecients in W. We assume the ordinarity of F: F|Dp = p 0 p with p 6= p ,p|Ip = N k 1 and p(Ip )=1

Arithmetic of the splitting field of Alexander polynomial

Abstract: In this paper, we study the arithmetic of the minimal splitting field of the Alexander polynomial of a knot and present two kinds of infinite families of knots, one being a family of knots which satisfy Heilbronn conjecture and the other a family of counterexamples to the conjecture.

A note on integral points on elliptic curves

Abstract: We investigate a problem considered by Zagier and Elkies, of finding large integral points on elliptic curves. By writing down a generic polynomial solution and equating coefficients, we are led to suspect four extremal cases that still might have nondegenerate solutions. Each of these cases gives rise to a polynomial system of equations, the first being solved by Elkies in 1988 using the resultant methods of~\Macsyma, with there being a unique rational nondegenerate solution. For the second case we found that resultants and/or Gr\"obner bases were not very efficacious. Instead, at the suggestion of Elkies, we used multidimensional $p$-adic Newton iteration, and were able to find a nondegenerate solution, albeit over a quartic number field. Due to our methodology, we do not have much hope of proving that there are no other solutions. For the third case we found a solution in a nonic number field, but we were unable to make much progress with the fourth case. We make a few concluding comments and include an appendix from Elkies regarding his calculations and correspondence with Zagier.

On algebras arising from the elements of a galois group for a galois algebra

Abstract: Let B be a ring with 1 and C the center of B It is shown that if B is a Galois algebra over R with a finite Galois group G, Jg = {b ∈ B |bx = g(x)b for all x ∈ B} for each g ∈ G, and eg an idempotent in C such that BJg = Beg, then the algebra B(g) generated by {Jh |h ∈ G and eh = eg} for an g ∈ G is a separable algebra over Reg and a central weakly Galois algebra with Galois group K(g) generated by {h ∈ G |eh = eg} Moreover, {B(g) |g ∈ G} and {K(g) |g ∈ G} are in a one-to-one correspondence, and three characterizations of a Galois extension are also given