Showing papers on "Generic polynomial published in 2007"
TL;DR: The Witt-Tits index of a loop torsor has been shown to be almost commutative in the case of almost commuting families of elements of finite order.
Abstract: 3 Loop torsors 8 3.1 The Witt-Tits index of a loop torsor . . . . . . . . . . . . . . . . . . . . . . 9 3.2 Reducibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.3 Almost commutative subgroups . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.4 Almost commuting families of elements of finite order . . . . . . . . . . . . 16 3.5 Almost commuting pairs and their invariants . . . . . . . . . . . . . . . . . 17 3.6 Failure in the anisotropic case . . . . . . . . . . . . . . . . . . . . . . . . . . 24
66 citations
TL;DR: In this article, the authors studied the Galois tower generated by iterates of a quadratic polynomial f defined over an arbitrary field, and defined a stochastic process associated to the tower that encodes root-fixing information at each level.
Abstract: We study the Galois tower generated by iterates of a quadratic polynomial f defined over an arbitrary field. One question of interest is to find the proportion an of elements at level n that fix at least one root; in the global field case these correspond to unramified primes in the base field that have a divisor at level n of residue class degree one. We thus define a stochastic process associated to the tower that encodes root-fixing information at each level. We develop a uniqueness result for certain permutation groups, and use this to show that for many f each level of the tower contains a certain central involution. It follows that the associated stochastic process is a martingale, and convergence theorems then allow us to establish a criterion for showing that an tends to 0. As an application, we study the dynamics of the family x 2 + c ∈ Fp[x ], and this in turn is used to establish a basic property of the p-adic Mandelbrot set.
41 citations
TL;DR: In this article, the Galois group of an abelian algebraic group over a global field was analyzed for several classes of elliptic curves and one-dimensional tori, and the authors gave a simple characterization of when the group is as large as possible up to constraints imposed by the endomorphism ring or Weil pairing.
Abstract: Given an abelian algebraic group $A$ over a global field $F$, $\alpha \in A(F)$, and a prime $\ell$, the set of all preimages of $\alpha$ under some iterate of $[\ell]$ generates an extension of $F$ that contains all $\ell$-power torsion points as well as a Kummer-type extension. We analyze the Galois group of this extension, and for several classes of $A$ we give a simple characterization of when the Galois group is as large as possible up to constraints imposed by the endomorphism ring or the Weil pairing. This Galois group encodes information about the density of primes $\p$ in the ring of integers of $F$ such that the order of $(\alpha \bmod{\p})$ is prime to $\ell$. We compute this density in the general case for several classes of $A$, including elliptic curves and one-dimensional tori. For example, if $F$ is a number field, $A/F$ is an elliptic curve with surjective 2-adic representation and $\alpha \in A(F)$ with $\alpha
ot\in 2A(F(A[4]))$, then the density of $\mathfrak{p}$ with ($\alpha \bmod{\p}$) having odd order is 11/21.
31 citations
TL;DR: In this paper, it was shown that the action of the absolute Galois group on dessins d'enfants of given genus g is faithful, a result that had been previously established for g = 0 and g = 1.
Abstract: We show that the action of the absolute Galois group on dessins d’enfants of given genus g is faithful, a result that had been previously established for g = 0 and g = 1.
22 citations
TL;DR: In this article, the problem of identifying the set K (G, Ω ) of Galois number fields with given Galois group G and root discriminant less than the Serre constant Ω ≈ 44.7632 was studied.
19 citations
TL;DR: Using class field theory, this work gets bounds for the order of the 4-torsion on J"X(F"q), the group of points defined over F"q on the Jacobian of a hyperelliptic curve X/F" q.
19 citations
TL;DR: In this article, it was shown that the number of equivalence classes of regular embeddings of a group G into InHol (G ) is equal to twice the total number of fixed point free endomorphisms of G.
19 citations
TL;DR: In this article, the kernel and cokernel of the S-capitulation map for arbitrary finite Galois extensions K/F (with Galois group G) and arbitrary finite sets of primes S of F (assumed to contain the archimedean primes in the number field case) were studied.
Abstract: This paper presents results on both the kernel and cokernel of the S-capitulation map C_{F,S}\ra C_{K,S}^{G} for arbitrary finite Galois extensions K/F (with Galois group G) and arbitrary finite sets of primes S of F (assumed to contain the archimedean primes in the number field case)
16 citations
01 Mar 2007
TL;DR: In this article, the authors studied the field isomorphism problem for a cubic generic polynomial X 3 + sX + s via Tschirnhausen transformation and showed that the fixed field under the action of the transformation is purely transcendental over an arbitrary base field.
Abstract: We study the field isomorphism problem for a cubic generic polynomial X 3 + sX + s via Tschirnhausen transformation. Through this process, there naturally appears a 2-dimensional involutive Cremona transformation. We show that the fixed field under the action of the transformation is purely transcendental over an arbitrary base field.
14 citations
01 Nov 2007
TL;DR: In this article, it was shown that for p > 5, the number of Hopf Galois structures on L/K afforded by K-Hopf algebras with associated group G is greater than p s, where s = (m-1) 2 3-m.
Abstract: Let p be an odd prime, G =Z m p , the elementary abelian p-group of rank m, and let Γ be the group of principal units of the ring Fp [x]/(x m+1 ). If L/K is a Galois extension with Galois group Γ, then we show that for p > 5, the number of Hopf Galois structures on L/K afforded by K-Hopf algebras with associated group G is greater than p s , where s = (m-1) 2 3- m.
12 citations
TL;DR: In this paper, it was shown that Noether's problem has an affirmative answer for the group GL(2, 3), over every field K. In particular, the group admits a generic polynomial over K.
Abstract: We prove that Noether’s problem has an affirmative answer for the group GL(2, 3), over every field K. In particular, the group $$\widetilde{S_4}\cong {\rm GL}(2,3)$$
admits a generic polynomial over $$\mathbb Q$$
. As a consequence, so does the group $$\widetilde{S_5}$$
.
TL;DR: In this article, the cohomological dimension of the maximal pro-p-quotient G of the absolute Galois group of F is at most n if and only if the corestriction maps Hn(H,{mathbb F}p) \to Gn(G,{Mathbb F}) are surjective for all open subgroups H of index p.
Abstract: Let p be a prime and F a field containing a primitive p-th root of unity. Then for n \in {mathbb N}, the cohomological dimension of the maximal pro-p-quotient G of the absolute Galois group of F is at most n if and only if the corestriction maps Hn(H,{mathbb F}p) \to Hn(G,{mathbb F}p) are surjective for all open subgroups H of index p. Using this result, we generalize Schreier's formula for dim{mathbb F}p H1(H,{mathbb F}p) to dim{mathbb F}p Hn(H, {mathbb F}p).
TL;DR: In this article, it was shown that R (M ) is the subgroup Cl ○ (M) of Cl (M), provided that the class number of k is odd, and that R(M) is the set of realizable classes such that there exists a Galois extension N/k which is tame, with Galois group isomorphic to Γ, and for which [ M ⊗ O k [ Γ ] O N ] = c.
TL;DR: In this article, the authors consider families of Laurent polynomials f 1, …, f n with a finite set of common zeros Z f in the torus T n = (C − { 0 } ) n, and present a new symbolic algorithm for computing the global residue as a rational function of the coefficients of the f i when the Newton polytopes of the F i are full-dimensional.
01 Jan 2007
TL;DR: In this paper, the results on permutation groups are applied to Galois theory to compute rapidly relations among roots of an univariate polynomial, in particular, to compute the relation between roots of a polynomials.
Abstract: Results on permutation groups are applyed to Galois theory In particular, to compute rapidly relations among roots of an univariate polynomial
27 Dec 2007
TL;DR: In this article, it was shown that there is a Picard-Vessiot extension e ⊃F for a matrix equation X' = XA(Y ij ), with differential Galois group SO n, with the property that if F is any differential field with field of constants C, then there is an extension E D F with differential GAs H < SO n if and only if there are fij ∈ F with A(f ij ) well defined and the equation X'' = Xa(fij) giving rise to the extension E �
Abstract: Let C be an algebraically closed field with trivial derivation and let F denote the differential rational field C , with Y ij , 1 < i < n - 1, 1 ≤ j ≤ n, i < j, differentially independent indeterminates over C. We show that there is a Picard-Vessiot extension e ⊃F for a matrix equation X' = XA.(Y ij ), with differential Galois group SO n , with the property that if F is any differential field with field of constants C, then there is a Picard-Vessiot extension E D F with differential Galois group H < SO n if and only if there are fij ∈ F with A(f ij ) well defined and the equation X' = XA(fij) giving rise to the extension E ⊃ F.
TL;DR: In this paper, the Galois group G = G S (p) of the maximal p-extension of Q unramified outside of S is mild when |S | = 4 and the cup product H 1 (G, Z/p Z ) ⊗ H 1(G, Z / p Z ) → H 2 ( G, Z, p Z ), is surjective.
TL;DR: In this paper, the authors provide an explicit description of the twisted Lie algebras of PGL3-equivariant derivations on the coordinate rings of F -irreducible PGL 3-torsors in terms of nine-dimensional central simple algesbras over F.
TL;DR: In this paper, it was shown that every tamely ramified Galois extension of K with Galois group isomorphic to G has a normal integral basis if and only if for every Galois extensions L/K with G, the ring of integers OL in L is free as a module over the associated order AL/K.
Abstract: Let p be a rational prime, G a group of order p, and K a number field containing a primitive pth root of unity. We show that every tamely ramified Galois extension of K with Galois group isomorphic to G has a normal integral basis if and only if for every Galois extension L/K with Galois group isomorphic to G, the ring of integers OL in L is free as a module over the associated order AL/K. We also give examples, some of which show that this result can still hold without the assumption that K contains a primitive pth root of unity.
25 Jul 2007
TL;DR: It is proved that the absolute value of every coefficient of f -- f is || f --∞ with at most one exception and the problem is reduced to solving systems of algebraic equations.
Abstract: For a real univariate polynomial f and a bounded closed domain D ⊂ C whose boundary C is a simple closed curve of finite length and is represented by a piecewise rational function, we provide a rigorous method for finding the real univariate polynomial f such that f has a zero in D and ||f -- f||∞ is minimal. First, we prove that the absolute value of every coefficient of f -- f is ||f -- f∞ with at most one exception. Using this property and the representation of C, we reduce the problem to solving systems of algebraic equations, each of which consists of two equations with two variables. Furthermore, every equation is of degree one with respect to one of the two variables.
TL;DR: In this article, it was shown that polynomial vector fields are not integrable with respect to their first integrals, or even of Darboux polynomials.
01 Jan 2007
TL;DR: In this article, it was shown that the number of degree k polynomials with coefficients in F that commute with f (under composition) is either zero or equal to the number one poly-nominals with coefficients of degree strictly greater than one in F (x).
Abstract: Let F be an algebraically closed field of characteristic 0 and f (x) a polynomial of degree strictly greater than one in F (x). We show that the number of degree k polynomials with coefficients in F that commute with f (under composition) is either zero or equal to the number of degree one poly- nomials with coefficients in F that commute with f . As a corollary, we obtain a theorem of E. A. Bertram characterizing those polynomials commuting with a Chebyshev polynomial.
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TL;DR: A geometric interpretation and generalisation for the Galois action on finite group character tables is sketched in this paper, where the generalisation is based on the Map_G(G^n,\bar{Q})/S_n for each finite G, where G acts by simultaneous conjugation on the n-tuples G^n and the symmetric group S_n permutes the components
Abstract: A geometric interpretation and generalisation for the Galois action on finite group character tables is sketched The generalisation is a Galois action on the space Map_G(G^n,\bar{Q})/S_n for each finite G, where G acts by simultaneous conjugation on the n-tuples G^n and the symmetric group S_n permutes the components
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TL;DR: The Galois group of irreducible polynomials with complex roots was shown to be isomorphic to a Frobenius group of degree 4k+1 in this paper.
Abstract: Let $f$ be an irreducible polynomial of prime degree $p\geq 5$ over $\QQ$, with precisely $k$ pairs of complex roots. Using a result of Jens H\"{o}chsmann (1999), we show that if $p\geq 4k+1$ then $\Gal(f/\QQ)$ is isomorphic to $A_{p}$ or $S_{p}$. This improves the algorithm for computing the Galois group of an irreducible polynomial of prime degree, introduced by A. Bialostocki and T.Shaska.
If such a polynomial $f$ is solvable by radicals then its Galois group is a Frobenius group of degree p. Conversely, any Frobenius group of degree p and of even order, can be realized as the Galois group of an irreducible polynomial of degree $p$ over $\QQ$ having complex roots.
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TL;DR: In this paper, an upper bound of the number of limit cycles which bifurcate from the period annulus of W under a generic polynomial perturbation was given.
Abstract: Let W be a weight-homogeneous planar polynomial differential system with a center. We find an upper bound of the number of limit cycles which bifurcate from the period annulus of W under a generic polynomial perturbation. We apply this result to a particular family of planar polynomial systems having a nilpotent center without meromorphic first integral.
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TL;DR: In this paper, an explicit polynomial in Q[x] that has Galois group SL2(F16) is presented, filling in a gap in the tables of Juergen Klueners and Gunther Malle.
Abstract: In this paper we show an explicit polynomial in Q[x] that has Galois group SL2(F16), filling in a gap in the tables of Juergen Klueners and Gunther Malle. The computation of this polynomial uses modular forms and their Galois representations.
01 Dec 2007
TL;DR: In this paper, it was shown that a generic polynomial foliation of the complex plane is topologically equivalent to only finitely many foliations, which is the weakest form of topological rigidity.
Abstract: Polynomial foliations of the complex plane are topologically rigid. Roughly speaking, this means that the topological equivalence of two foliations implies their affine equivalence. There exist various nonequivalent formalizations of the notion of topological rigidity. Generic polynomial foliations of fixed degree have the so-called property of absolute rigidity, which is the weakest form of topological rigidity. This property was discovered by the author more than 30 years ago. The genericity conditions imposed at that time were very restrictive. Since then, this topic has been studied by Shcherbakov, Gomez-Mont, Nakai, Lins Neto-Sad-Scardua, Loray-Rebelo, and others. They relaxed the genericity conditions and increased the dimension. The main conjecture in this field states that a generic polynomial foliation of the complex plane is topologically equivalent to only finitely many foliations. The main result of this paper is weaker than this conjecture but also makes it possible to compare topological types of distant foliations.
TL;DR: In this paper, the relation between Galois groups and Galois ensembles was described, where S is a submonoid of (the set of all natural numbers) and G is an associative group of inverse polynomial modules.
Abstract: Given an injective envelope E of a left R-module M, there is an associative Galois group Gal. Let R be a left noetherian ring and E be an injective envelope of M, then there is an injective envelope of an inverse polynomial module as a left R[x]-module and we can define an associative Galois group Gal. In this paper we describe the relations between Gal and Gal. Then we extend the Galois group of inverse polynomial module and can get Gal, where S is a submonoid of (the set of all natural numbers).
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TL;DR: In this paper, it was shown that for every splitting of a polynomial with noncommutative coefficients into linear factors, any cyclic permutation of linear factors gives the same result and all $a_{k}$ are roots of that polynomials.
Abstract: It is shown that for every splitting of a polynomial with noncommutative coefficients into linear factors $(X-a_{k})$ with $a_{k}$'s commuting with coefficients, any cyclic permutation of linear factors gives the same result and all $a_{k}$ are roots of that polynomial. Examples are given and analyzed from Galois theory point of view.
TL;DR: In this article, a new proof of self-duality for Selmer groups is given, showing that if A is an abelian variety over a number field K, and F/K is a Galois extension with Galois group G, then the Q_pG-representation naturally associated to the p-infinity Selmer group of A/F is selfdual.
Abstract: The first part of the paper gives a new proof of self-duality for Selmer groups: if A is an abelian variety over a number field K, and F/K is a Galois extension with Galois group G, then the Q_pG-representation naturally associated to the p-infinity Selmer group of A/F is self-dual. The second part describes a method for obtaining information about parities of Selmer ranks from the local Tamagawa numbers of A in intermediate extensions of F/K.