Showing papers on "Generic polynomial published in 2000"

Rational connectedness and Galois covers of the projective line

Abstract: Let k be a p-adic field. Some time ago, D. Harbater [9] proved that anyfinite group G may be realized as a regular Galois group over the rationalfunction field in one variable k(t), namely there exists a finite field extensionF/k(t), Galois with group G, such that F is a regular extension of k (i.e. kis algebraically closed in F). Moreover, one may arrange that a given k-placeof k(t) be totally split in F. Harbater proved this theorem for k an arbitrarycomplete valued field. Rather formal arguments ([10, §4.5]; §2 hereafter) thenimply that the theorem holds over any ‘large’ field k. This in turn is a specialcase of a result of Pop [15], hence will be referred to as the Harbater/Poptheorem. We refer to [10], [16], [6] for precise references to the literature (workof D`ebes, Deschamps, Fried, Haran, Harbater, Jarden, Liu, Pop, Serre, andV¨olklein).Most proofs (see [10], [19, 8.4.4, p. 93] and Liu’s contribution to [16]; seehowever [15]) first use direct arguments to establish the theorem when G is acyclic group (here the nature of the ground field is irrelevant), then proceed bypatching, using either formal or rigid geometry, together with GAGA theorems.In the present paper, where I take the case of algebraically closed fieldsfor granted, I show how a technique recently developed by Kolla´r [12] may beused to give a quite different proof of the Harbater/Pop theorem, when the‘large’ field k has characteristic zero. This proof actually gives more than theoriginal result (see comment after statement of Theorem 1).Before I formally state the main result, let us recall what a ‘large’ field is.Let k be a field and let k((y)) be the quotient field of the ring k[[y]] of formalpower series in one variable. Following F. Pop, we shall say that k is ‘large’ ifit satisfies one of the three equivalent properties ([15, Prop. 1.1]):(i) It is existentially closed in k((y)): any k-variety with a k((y))-point hasa k-point.(ii) On a smooth integral k-variety with a k-point, k-points are Zariski dense.(iii) On a smooth integral k-curve with a k-point, k-points are Zariski dense.

Some polynomials over Q ( t ) and their Galois groups

Abstract: Examples of polynomials with Galois group over Q(t) corresponding to every transitive group through degree eight are calculated, constructively demonstrating the existence of an infinity of extensions with each Galois group over Q through degree eight. The methods used, which for the most part have not appeared in print, are briefly discussed.

Galois Groups Over Nonrigid Fields

Abstract: Let F be a field with char F 6 2. We show that F is a nonrigid field if and only if certain small 2-groups occur as Galois groups over F. These results provide new "automatic realizability" results for Galois groups over F. The groups we consider demonstrate the inequality of two particular metabelian 2-extensions of F which are unequal precisely when F is a nonrigid field. Using known results on connections between rigidity and existence of certain valuations, we obtain Galois-theoretic criteria for the existence of these valuations.

The dynamics of linearized polynomials

Abstract: Let F = GF(q). To any polynomial G element of F[x] there is associated a mapping on the set I_F of monic irreducible polynomials over F. We present a natural and effective theory of the dynamics of the mapping for the case in which G is a monic q-linearized polynomial. The main outcome is the following theorem. Assume that G is not of the form x^(q^l), where l>= 0 (in which event the dynamics is trivial). Then, for every integer n >= 1 and for every integer k >= 0, there exist infinitely many mu element of I_F having preperiod k and primitive period n with respect to the mapping. Previously, Morton, by somewhat different means, had studied the primitive periods of the mapping when G = x^q - ax, a a non-zero element of F. Our theorem extends and generalizes Morton's result. Moreover, it establishes a conjecture of Morton for the class of q-linearized polynomials.

Universal normal bases for the abelian closure of the field of rational numbers

Abstract: If E/F is a finite-dimensional Galois extension with Galois group G, then, by the Normal Basis Theorem, there exist elements w of E such that {g(w) | g element of G} is an F-basis of E, a so-called normal basis, whence w is called normal in E/F. In the present Paper, we study normal bases for cyclotomic fields.

On characterizations of a center galois extension

Abstract: Let B be a ring with 1, C the center of B, G a finite automorphism group of B, and B G the set of elements in B fixed under each element in G. Then, it is shown that B is a center Galois extension of B G (that is, C is a Galois algebra over C G with Galois group G|CG) if and only if the ideal of B generated by{c − g(c) | c ∈ C} is B for each g ≠ 1i n G. This generalizes the well known characterization of a commutative Galois extension C that C is a Galois extension of C G with Galois group G if and only if the ideal generated by {c − g(c) | c ∈ C} is C for each g ≠ 1i nG. Some more characterizations of a center Galois extension B are also given.

Noncommutative algebras associated to complexes and graphs

Abstract: This is a first of our papers devoted to "noncommutative topology and graph theory". Its origin is the paper math.QA/0002238 by I. Gelfand, V. Retakh, and R.L. Wilson where a new class of noncommutative algebras $Q_n$ was introduced. The algebra $Q_n$ is closely related to factorizations of a generic polynomial of degree $n$ over a division algebra into linear factors.

The general ikehata theorem for h-separable crossed products

Abstract: Let B be a ring with 1, C the center of B, G an automorphism group of B of order n for some integer n, C G the set of elements in C fixed under G, ∆ = ∆ (B, G,f) a crossed product over B where f is a factor set from G × G to U(C G ). It is shown that ∆ is an H-separable extension of B and V∆(B) is a commutative subring of ∆ if and only if C is a Galois algebra over C G with Galois group G|CG.

On generic cyclic polynomials of odd prime degree

Abstract: Using Cohen's construction of defining polynomials for a cyclic group of odd prime order, we define a polynomial with some parameters which generates cyclic extensions of a given odd prime degree, and prove it to be generic in the sense as defined below.

Computing the Galois group of a polynomial using linear differential equations

Abstract: In this paper we show how to compute the Galois group G of a polynomial ƒ ∈ Q(x)[Y] by factoring the associated linear differential equation Lƒ(Y) = 0 (and constructions of it) of minimal order satisfied by the roots of ƒ. We use that the differential Galois group of Lƒ(Y) is a faithful linear representation of G whose character is a summand of the permutation character of G acting on the roots of ƒ. Our approach is motivated by the fact that the orders of the involved differential equations are much lower than the degrees of the Lagrange resolvants of ƒ. In the final section we show how, if ƒ ∈ Q(x)[Y], our approach via differential Galois theory helps one to also compute the Galois group of ƒ over Q(x).

Free product of absolute Galois groups

Abstract: The free profinite product of finitely many absolute Galois group is an absolute Galois group

On the Hensel lift of a polynomial

Abstract: Denote by R the Galois ring of characteristic p/sup e/ and cardinality p/sup em/, where p is a prime and e and m are positive integers. Let g(x) be a monic polynomial over F/sub p/m. A polynomial f(x) over R is defined to be a Hensel lift of g(x) in R[x] if f~(x)=g(x), and there is a positive integer n not divisible by p such that f(x) divides x/sup n/-1 in R[x]. It is proved that g(x) has a unique Hensel lift in R[x] if and only if g(x) has no multiple roots and x/spl chi/g(x). An algorithm to compute the Hensel lift is also given.

A method for deciding whether the Galois group is abelian

Abstract: We propose a polynomial time algorithm to decide whether the Galois group of an irreducible polynomial ƒ ∈ Q[x] is abelian, and, if so, determine all its elements along with their action on the set of roots of ƒ. This algorithm does not require factorization of polynomials over number fields. Instead we shall use the quadratic Newton—Lifting and the truncated expressions of the roots of ƒ over a p—adic number field Qp, for an appropriate prime p in Z.

A condition for an area-preserving mapping to be in the Engel’s form

Abstract: We establish explicit expressions of restrictions on the coefficients of nonlinear terms in a two-dimensional area-preserving polynomial map imposed by the property of area preserving. We also establish a necessary and sufficient condition for a two-dimensional area-preserving generic polynomial map to be in the Engel’s form. The condition is that coefficients of nonlinear terms in the first mapping equation are proportional to the corresponding ones in the second mapping equation. As an application of the results, we discuss how to regain the area-preserving property lost in truncating a two-dimensional area-preserving map.