5,078 research outputs found

    The set of stable primes for polynomial sequences with large Galois group

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    Let KK be a number field with ring of integers OK\mathcal O_K, and let {fk}k∈N⊆OK[x]\{f_k\}_{k\in \mathbb N}\subseteq \mathcal O_K[x] be a sequence of monic polynomials such that for every n∈Nn\in \mathbb N, the composition f(n)=f1∘f2∘…∘fnf^{(n)}=f_1\circ f_2\circ\ldots\circ f_n is irreducible. In this paper we show that if the size of the Galois group of f(n)f^{(n)} is large enough (in a precise sense) as a function of nn, then the set of primes p⊆OK\mathfrak p\subseteq\mathcal O_K such that every f(n)f^{(n)} is irreducible modulo p\mathfrak p has density zero. Moreover, we prove that the subset of polynomial sequences such that the Galois group of f(n)f^{(n)} is large enough has density 1, in an appropriate sense, within the set of all polynomial sequences.Comment: Comments are welcome

    Polynomials with prescribed bad primes

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    We tabulate polynomials in Z[t] with a given factorization partition, bad reduction entirely within a given set of primes, and satisfying auxiliary conditions associated to 0, 1, and infinity. We explain how these sets of polynomials are of particular interest because of their role in the construction of nonsolvable number fields of arbitrarily large degree and bounded ramification. Finally we discuss the similar but technically more complicated tabulation problem corresponding to removing the auxiliary conditions.Comment: 26 pages, 3 figure

    A Survey of Finite Algebraic Geometrical Structures Underlying Mutually Unbiased Quantum Measurements

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    The basic methods of constructing the sets of mutually unbiased bases in the Hilbert space of an arbitrary finite dimension are discussed and an emerging link between them is outlined. It is shown that these methods employ a wide range of important mathematical concepts like, e.g., Fourier transforms, Galois fields and rings, finite and related projective geometries, and entanglement, to mention a few. Some applications of the theory to quantum information tasks are also mentioned.Comment: 20 pages, 1 figure to appear in Foundations of Physics, Nov. 2006 two more references adde

    Visibly irreducible polynomials over finite fields

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    H. Lenstra has pointed out that a cubic polynomial of the form (x-a)(x-b)(x-c) + r(x-d)(x-e), where {a,b,c,d,e} is some permutation of {0,1,2,3,4}, is irreducible modulo 5 because every possible linear factor divides one summand but not the other. We classify polynomials over finite fields that admit an irreducibility proof with this structure.Comment: 11 pages. To appear in the American Mathematical Monthl

    Algorithms in algebraic number theory

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    In this paper we discuss the basic problems of algorithmic algebraic number theory. The emphasis is on aspects that are of interest from a purely mathematical point of view, and practical issues are largely disregarded. We describe what has been done and, more importantly, what remains to be done in the area. We hope to show that the study of algorithms not only increases our understanding of algebraic number fields but also stimulates our curiosity about them. The discussion is concentrated of three topics: the determination of Galois groups, the determination of the ring of integers of an algebraic number field, and the computation of the group of units and the class group of that ring of integers.Comment: 34 page

    Computation of Integral Bases

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    Let AA be a Dedekind domain, KK the fraction field of AA, and f∈A[x]f\in A[x] a monic irreducible separable polynomial. For a given non-zero prime ideal p\mathfrak{p} of AA we present in this paper a new method to compute a p\mathfrak{p}-integral basis of the extension of KK determined by ff. Our method is based on the use of simple multipliers that can be constructed with the data that occurs along the flow of the Montes Algorithm. Our construction of a p\mathfrak{p}-integral basis is significantly faster than the similar approach from [7][7] and provides in many cases a priori a triangular basis.Comment: 22 pages, 4 figure

    Commutative association schemes

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    Association schemes were originally introduced by Bose and his co-workers in the design of statistical experiments. Since that point of inception, the concept has proved useful in the study of group actions, in algebraic graph theory, in algebraic coding theory, and in areas as far afield as knot theory and numerical integration. This branch of the theory, viewed in this collection of surveys as the "commutative case," has seen significant activity in the last few decades. The goal of the present survey is to discuss the most important new developments in several directions, including Gelfand pairs, cometric association schemes, Delsarte Theory, spin models and the semidefinite programming technique. The narrative follows a thread through this list of topics, this being the contrast between combinatorial symmetry and group-theoretic symmetry, culminating in Schrijver's SDP bound for binary codes (based on group actions) and its connection to the Terwilliger algebra (based on combinatorial symmetry). We propose this new role of the Terwilliger algebra in Delsarte Theory as a central topic for future work.Comment: 36 page
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