Showing papers on "Generic polynomial published in 2012"
TL;DR: For a prime power q = pd and a field F containing a root of unity of order q, this article showed that the Galois cohomology ring is determined by a quotient of the absolute Galois group GF related to its descending q-central sequence.
Abstract: For a prime power q = pd and a field F containing a root of unity of order q we show that the Galois cohomology ring \({H^*(G_F,\mathbb{Z}/q)}\) is determined by a quotient \({G_F^{[3]}}\) of the absolute Galois group GF related to its descending q-central sequence. Conversely, we show that \({G_F^{[3]}}\) is determined by the lower cohomology of GF. This is used to give new examples of pro-p groups which do not occur as absolute Galois groups of fields.
70 citations
12 Mar 2012
TL;DR: This paper addresses the problem of formal verification of hardware implementations of (modulo) multipliers over Galois fields of the type F(2k), using a computer-algebra/algebraic-geometry based approach and proves that this term order renders the set of polynomials itself a Gröbner basis of this ideal - thus significantly improving verification.
Abstract: Galois field arithmetic finds application in many areas, such as cryptography, error correction codes, signal processing, etc Multiplication lies at the core of most Galois field computations This paper addresses the problem of formal verification of hardware implementations of (modulo) multipliers over Galois fields of the type F 2k , using a computer-algebra/algebraic-geometry based approach The multiplier circuit is modeled as a polynomial system in F 2k [x 1 , x 2 , … , x d ] and the verification problem is formulated as a membership test in a corresponding (radical) ideal This requires the computation of a Grobner basis, which can be computationally intensive To overcome this limitation, we analyze the circuit topology and derive a term order to represent the polynomials Subsequently, using the theory of Grobner bases over Galois fields, we prove that this term order renders the set of polynomials itself a Grobner basis of this ideal - thus significantly improving verification Using our approach, we can verify the correctness of, and detect bugs in, upto 163-bit circuits in F 2 163 ; whereas contemporary approaches are infeasible
26 citations
TL;DR: Inverse Galois Theory as mentioned in this paper showed that for every positive integer m ≡ 2 ( mod 4 ), there is a Laguerre polynomial of degree m with associated Galois group A m.
24 citations
TL;DR: In this paper, it was shown that a very small canonical Galois group over a field containing a root of unity of order (G_F) is sufficient to encode the valuations on a field whose value group is not $p$-divisible and which satisfies a variant of Hensel's lemma.
Abstract: Let $p$ be a prime number and let $F$ be a field containing a root of unity of order $p$. We prove that a certain very small canonical Galois group $(G_F)_{[3]}$ over $F$ encodes the valuations on $F$ whose value group is not $p$-divisible and which satisfy a variant of Hensel's lemma.
20 citations
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TL;DR: The parameterized generic Galois group of a q-difference module was introduced in this article, which is a differential group associated to the smallest differential tannakian category generated by the q-Difference module, equipped with the forgetful functor.
Abstract: We introduce the parameterized generic Galois group of a q-difference module, that is a differential group in the sense of Kolchin. It is associated to the smallest differential tannakian category generated by the q-difference module, equipped with the forgetful functor. Our previous results on the Grothendieck conjecture for q-difference equations lead to an adelic description of the parameterized generic Galois group, in the spirit of the Grothendieck-Katz's conjecture on p-curvatures. Using this description, we show that the Malgrange-Granier D-groupoid of a nonlinear q-difference system coincides, in the linear case, with the parameterized generic Galois group introduced here. The paper is followed by an appendix by A. Granier, that provides a quick introduction to the D-groupoid of a non-linear q-difference equation.
13 citations
Proceedings Article•
11 Oct 2012
TL;DR: A new efficient algorithm for building a Galois sub-hierarchy which runs in O(min{nm, n α }), where n is the number of objects or attributes, m is the size of the relation, and n α is the time required to perform matrix multiplication.
Abstract: Given a relation R ⊆ O × A on a set O of objects and a set A of attributes, the Galois sub-hierarchy (also called AOC-poset) is the partial order on the introducers of objects and attributes in the corresponding concept lattice. We present a new efficient algorithm for building a Galois sub-hierarchy which runs in O(min{nm, n α }), where n is the number of objects or attributes, m is the size of the relation, and n α is the time required to perform matrix multiplication (currently α = 2.376).
11 citations
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TL;DR: In this article, a q-analogue of the Grothendieck conjecture on p-curvatures for q-difference equations defined over the field of rational functions K(x), where K is a finite extension of a rational function k(q), with k perfect, was given.
Abstract: In the present paper, we give a q-analogue of the Grothendieck conjecture on p-curvatures for q-difference equations defined over the field of rational function K(x), where K is a finite extension of a field of rational functions k(q), with k perfect. Then we consider the generic (also called intrinsic) Galois group in the sense of N. Katz. The result in the first part of the paper lead to a description of the generic Galois group through the properties of the functional equations obtained specializing q on roots of unity. Although no general Galois correspondence holds in this setting, in the case of positive characteristic, where nonreduced groups appear, we can prove some devissage of the generic Galois group. In the last part of the paper, we give a complete answer to the analogue of Grothendieck conjecture on $p$-curvatures for q-difference equations defined over the field of rational function K(x), where K is any finitely generated extension of \mathbb Q and q
eq 0,1: we prove that the generic Galois group of a q-difference module over K(x) always admits an adelic description in the spirit of the Grothendieck-Katz conjecture. To this purpose, if q is an algebraic number, we prove a generalization of the results by L. Di Vizio, 2002.
10 citations
Posted Content•
TL;DR: Granboulan and Matzat as mentioned in this paper showed that there is a 1-parameter family of polynomials whose Galois group over the rational function field Q(t) is the Mathieu group M24.
Abstract: Granboulan computed an explicit polynomial whose Galois group over the rational function field Q(t) is the Mathieu group M24. By a result of Malle and Matzat, it was known before that such a polynomial exists. Even more, their proof showed that there is a 1-parameter family of such polynomials. In this note we compute this family.
5 citations
TL;DR: In this paper, it was shown that the characteristic variety of a generic polynomial vector field has no homoge-neous involutive subvarieties besides the zero section and fibers over singular points.
Abstract: We confirm a conjecture of Bernstein-Lunts which predicts that the characteristic variety of a generic polynomial vector field has no homoge- neous involutive subvarieties besides the zero section and subvarieties of fibers over singular points.
5 citations
TL;DR: In this paper, the authors extend this result to a larger class of polynomials, which leads to trinomials of degree n for each n with Galois group Sn, the symmetric group on n letters.
Abstract: A well known result of Schur states that if n is a positive integer and a0, a1,…,an are arbitrary integers with a0an coprime to n!, then the polynomial is irreducible over the field ℚ of rational numbers. In case each ai = 1, it is known that the Galois group of fn(x) over ℚ contains An, the alternating group on n letters. In this paper, we extend this result to a larger class of polynomials fn(x) which leads to the construction of trinomials of degree n for each n with Galois group Sn, the symmetric group on n letters.
3 citations
01 Jan 2012
TL;DR: In this article, the authors show two compositions of Galois extensions, one for B over B N, and another for H over (�H) N, where H is a Galois algebra with Galois group N/N 0 where N is the normalizer of H in G and N 0 = {g ∈ N, g(x) = x for all x ∈ �H }, and show that both compositions of these compositions can be expressed in terms of a separable C G −algebra.
Abstract: Let B be a ring with 1 and a Galois extension of B G with Galois group G such that B G is a separable C G −algebra where C is the center of B, H a subgroup of G andH the commutator subring of B H in B. IfH is a Galois algebra with Galois group N/N0 where N is the normalizer of H in G and N0 = {g ∈ N, g(x) = x for all x ∈ �H }, then we show two compositions of Galois extensions, one for B over B N , and another forH over (�H) N.
01 Jan 2012
TL;DR: In this article, a sufficient condition on a set with a maximal number of subgroups is given under which α is one-to-one on the set, and the collection of such sets of sub groups is computed, and thus we can determine which Galois group H is unique for the Galois extension B over B H.
Abstract: Let B be a Galois extension of B G with Galois group G, and α : H −→ B H the Galois map from the set of subgroups of G to the set of subextensions of B G . Then a sufficient condition on a set with a maximal number of subgroups is given under which α is one-to-one on the set. Moreover, the collection of such sets of subgroups is computed, and thus we can determine which Galois group H is unique for the Galois extension B over B H
TL;DR: In this paper, a general method for computing invariants of large degree was proposed, as well as various special invariants that are derived from the structure of the groups and applied to the task of computing the Galois groups of polynomials over the rational numbers.
Abstract: Computational Galois theory, in particular the problem of computing the Galois group of a given polynomial is a very old problem. Currently, the best algorithmic solution is Stauduhar's method. Computationally, one of the key challenges in the application of Stauduhar's method is to find, for a given pair of groups H
TL;DR: In this paper, the problem of describing linear maps ϕ : A → A that preserve zeros of a polynomial in an algebra is studied for general polynomials f in the case where A = M n (F ).
TL;DR: Using the method of exponential sums, Sidel’nikov obtained estimates of the frequencies of occurrence of elements on segments of linear recurrence sequences over finite fields, generalized to the case of Galois rings.
Abstract: Using the method of exponential sums, Sidel’nikov obtained estimates of the frequencies of occurrence of elements on segments of linear recurrence sequences over finite fields. These results are generalized to the case of Galois rings. It is shown that, in some cases, the estimates obtained in this paper are sharper than previously known ones.
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TL;DR: An algebraic technique for derivation of fast discrete cosine transform (DCT) algorithms and Galois theory is presented, which defines polynomial algebra over the field of rational numbers Q instead of complex C.
Abstract: The paper presents an algebraic technique for derivation of fast discrete cosine transform (DCT) algorithms. The technique is based on the algebraic signal processing theory (ASP). In ASP a DCT associates with a polynomial algebra C[x]/p(x). A fast algorithm is obtained as a stepwise decomposition of C[x]/p(x). In order to reveal the connection between derivation of fast DCT algorithms and Galois theory we define polynomial algebra over the field of rational numbers Q instead of complex C. The decomposition of Q[x]/p(x) requires the extension of the base field Q to splitting field E of polynomial p(x). Galois theory is used to find intermediate subfields L_i in which polynomial p(x) is factored. Based on this factorization fast DCT algorithm is derived.
Journal Article•
TL;DR: In this paper, a generic polynomial for the dihedral group D8 of order 16 in characteristic 2 was derived for a fixed field K(x1,..., xm), where φi is in K (x 1,.,.,, xi−1)[xi] and has minimal positive degree di in xi.
Abstract: Let K be a field and suppose that G is a finite group that acts faithfully on K(x1, . . . , xm) by automorphisms of the form g(xi) = ai(g)xi + bi(g), where ai(g), bi(g) ∈ K(x1, . . . , xi−1) for all g ∈ G and all i = 1, . . . ,m. As shown by Miyata, the fixed field K(x1, . . . , xm) G is purely transcendental over K and admits a transcendence basis {φ1, . . . , φm}, where φi is in K(x1, . . . , xi−1)[xi] and has minimal positive degree di in xi. We determine exactly the degree di of each invariant φi as a polynomial in xi and show the relation d1 · · · dm = |G|. As an application, we compute a generic polynomial for the dihedral group D8 of order 16 in characteristic 2. Acknowledgements: This research was conducted at Louisiana State University as part of the Research Experience for Undergraduates Program funded by National Science Foundation grant DMS-0648064. The author would like to thank Dr. Jorge Morales for his guidance. Page 54 RHIT Undergrad. Math. J., Vol. 13, No. 1
TL;DR: This work gives the first polynomial-time algorithm for checking whether the Galois group Gal(f) of an inputPolynomial ∈ Q[i>X
Abstract: We give the first polynomial-time algorithm for checking whether the Galois group Gal(f) of an input polynomial f(X) ∈ Q[X] is nilpotent: the running time of our algorithm is bounded by a polynomial in the size of the coefficients of f and the degree of f. Additionally, we give a deterministic polynomial-time algorithm that, when given as input a polynomial f(X) ∈ Q[X] with nilpotent Galois group, computes for each prime factor p of n Gal(f), a polynomial gp(X)∈ Q[X] whose Galois group of is the p-Sylow subgroup of Gal(f).
Dissertation•
14 May 2012
TL;DR: In this article, the Galois group for solvability for cubic, quartic and quintic polynomials has been studied and shown to be solvable by radicals.
Abstract: The solvability by radicals is shown through the use of Galois theory. General polynomial of degree five or more are not solvable and hence no general formulas exist. Here we study the Galois group for solvability for cubic, quartic and quintic polynomials. The Galois group has a wide physical application in the field of theoretical physics.
25 Dec 2012
TL;DR: In this article, it was shown that the set of all polynomial permutations of Z p r is a group and the structure of the group and related groups is known.
Abstract: . The set of all polynomial permutations of Z p r forms agroup. We investigate the structure of the group and some relatedgroups, and completely determine the structure of the group of allpolynomial permutations of Z p 2 . 1. IntroductionLet p r be a prime power. If a polynomial over the Galois ring Z p r induces a permutation of Z p r , then it is called a permutation polynomial.For r= 1, it is well-known that every permutation of the eld Z p isinduced by a polynomial [4]. On the other hand, for r>1, not everypermutation of Z p r is induced by a polynomial. Hence the notion of apolynomial permutation, that is, permutation induced by a polynomialis meaningful in this case.It is easy to see that the set of all polynomial permutations of Z p r isa group. Indeed the set of all polynomial permutations of Z p r is clearlyclosed under composition and is a nite subset of the symmetric groupof Z p r , and hence forms a subgroup. We investigate the structure ofthis group and related groups. In particular, we completely determinethe structure of the group of all polynomial permutations of Z
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TL;DR: In this article, the methods of classical invariant theory are used to construct generic polynomials for groups $S_5$ and $A_5, along with explicit reductions to specializations of the generic poynomials defining any desired field extension with those groups.
Abstract: The methods of classical invariant theory are used to construct generic polynomials for groups $S_5$ and $A_5$, along with explicit reductions to specializations of the generic polynomials defining any desired field extension with those groups.
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TL;DR: In this paper, a generic polynomial mapping mapping from the plane to the plane is considered, and there are constructed quadratic forms whose signatures determine the number of positive and negative cusps of the mapping.
Abstract: Let f be a generic polynomial mapping mapping from the plane to the plane. There are constructed quadratic forms whose signatures determine the number of positive and negative cusps of f.