Search

Your search keyword '"Bary-Soroker, Lior"' showing total 196 results
Start Over Author "Bary-Soroker, Lior"
196 results on '"Bary-Soroker, Lior"'

52. Projective pairs of profinite groups

Author

Bary-Soroker, Lior

Subjects
Mathematics - Group Theory, 20E18, 12E30
Abstract

We generalize the notion of a projective profinite group to a projective pair of a profinite group and a closed subgroup. We establish the connection with Pseudo Algebraically Closed (PAC) extensions of PAC fields: Let M be an algebraic extension of a PAC field K. Then M/K is PAC if and only if the corresponding pair of absolute Galois groups (Gal(M),Gal(K)) is projective. Moreover any projective pair can be realized as absolute Galois groups of a PAC extension of a PAC field. Using this characterization we construct new examples of PAC extensions of relatively small fields, e.g., unbounded abelian extensions of the rational numbers.

Published
2008

53. Permanence criteria for semi-free profinite groups

Author

Bary-Soroker, Lior, Haran, Dan, and Harbater, David

Subjects
Mathematics - Group Theory, Mathematics - Algebraic Geometry, 12E30, 12F10, 20E18, 14H30
Abstract

We introduce the condition of a profinite group being semi-free, which is more general than being free and more restrictive than being quasi-free. In particular, every projective semi-free profinite group is free. We prove that the usual permanence properties of free groups carry over to semi-free groups. Using this, we conclude that if k is a separably closed field, then many field extensions of k((x,y)) have free absolute Galois groups., Comment: 24 pages

Published
2008
Full Text
View/download PDF

54. On Pseudo Algebraically Closed Extensions of Fields

Author

Bary-Soroker, Lior

Subjects
Mathematics - Number Theory, 12E30, 12F10
Abstract

The notion of `Pseudo Algebraically Closed (PAC) extensions' is a generalization of the classical notion of PAC fields. It was originally motivated by Hilbert's tenth problem, and recently had new applications. In this work we develop a basic machinery to study PAC extensions. This machinery is based on a generalization of embedding problems to field extensions. The main goal is to prove that the Galois closure of any proper separable algebraic PAC extension is its separable closure. This vastly generalizes earlier works of Jarden-Razon, Jarden, and Jarden and the author. This also leads to a classification of all finite PAC extensions which in turn proves the `bottom conjecture' for finitely generated infinite fields. The secondary goal of this work is to unify proofs of known results about PAC extensions and to establish new basic properties of PAC extensions, e.g.\ transitiveness of PAC extensions., Comment: 31 pages, small corrections, change of commutative diagrams, to be published in the Journal of Algebra

Published
2008
Full Text
View/download PDF

55. On the characterization of Hilbertian fields

Author

Bary-Soroker, Lior

Subjects
Mathematics - Number Theory, 12E25
Abstract

The main goal of this work is to answer a question of P. D`ebes and D. Haran by relaxing the condition for Hilbertianity. Namely we prove that for a field K to be Hilbertian it suffices that K has the irreducible specialization property merely for absolutely irreducible polynomials., Comment: (v2) contains a new application of the main result (v3) Changed the numbering of thms, lemmas, etc., to fit the published version

Published
2008
Full Text
View/download PDF

56. On the remainder in the Taylor theorem

Author

Bary-Soroker, Lior and Leher, Eli

Subjects
Mathematics - Classical Analysis and ODEs, 26A24
Abstract

We give a short straightforward proof for the bound of the reminder term in the Taylor theorem. The proof uses only induction and the fact that $f'\geq 0$ implies the monotonicity of $f$, so it might be an attractive proof to give to undergraduate students., Comment: 2 pages, the proof was shortened

Published
2008

57. PAC Fields over Finitely Generated Fields

Author

Bary-Soroker, Lior and Jarden, Moshe

Subjects
Mathematics - Number Theory, 12E30
Abstract

We prove the following theorem for a finitely generated field $K$: Let $M$ be a Galois extension of $K$ which is not separably closed. Then $M$ is not PAC over $K$., Comment: 7 pages, Math. Z

Published
2007

58. On PAC Extensions and Scaled Trace Forms

Author

Bary-Soroker, Lior and Kelmer, Dubi

Subjects
Mathematics - Number Theory
Abstract

Any non-degenerate quadratic form over a Hilbertian field (e.g., a number field) is isomorphic to a scaled trace form. In this work we extend this result to more general fields. In particular, prosolvable and prime-to-p extensions of a Hilbertian field. The proofs are based on the theory of PAC extensions., Comment: 10 pages; added results on prime-to-p extensions, removed the condition on the countability of the fields

Published
2007

59. The Minimal Ramification Problem for Rational Function Fields over Finite Fields.

Author

Bary-Soroker, Lior, Entin, Alexei, and Fehm, Arno

Subjects
*FINITE groups, *PRIME numbers, *FINITE fields, *NUMBER theory
Abstract

We study the minimal number of ramified primes in Galois extensions of rational function fields over finite fields with prescribed finite Galois group. In particular, we obtain a general conjecture in analogy with the well studied case of number fields, which we establish for abelian, symmetric, and alternating groups in many cases. [ABSTRACT FROM AUTHOR]

Published
2023
Full Text
View/download PDF

63. Hilbert's Irreducibility Theorem via Random Walks.

Author

Bary-Soroker, Lior and Garzoni, Daniele

Subjects
*RANDOM walks, *LINEAR algebraic groups, *POINT set theory, *SEMISIMPLE Lie groups, *CAYLEY graphs
Abstract

Let |$G$| be a connected linear algebraic group over a number field |$K$|⁠ , let |$\Gamma $| be a finitely generated Zariski dense subgroup of |$G(K)$|⁠ , and let |$Z\subseteq G(K)$| be a thin set, in the sense of Serre. We prove that, if |$G/\textrm {R}_{u}(G)$| is either trivial or semisimple and |$Z$| satisfies certain necessary conditions, then a long random walk on a Cayley graph of |$\Gamma $| hits elements of |$Z$| with negligible probability. We deduce corollaries to Galois covers, characteristic polynomials, and fixed points in group actions. We also prove analogous results in the case where |$K$| is a global function field. [ABSTRACT FROM AUTHOR]

Published
2023
Full Text
View/download PDF

88. Chebotarev density theorem in short intervals for extensions of Fq(T).

Author

Bary-Soroker, Lior, Gorodetsky, Ofir, Karidi, Taelin, and Sawin, Will

Subjects
*NUMBER theory, *RIEMANN hypothesis, *PRIME numbers, *FINITE fields, *CONJUGACY classes, *ARITHMETIC functions
Abstract

An old open problem in number theory is whether the Chebotarev density theorem holds in short intervals. More precisely, given a Galois extension E of Q with Galois group G, a conjugacy class C in G, and a 1 > ε > 0, one wants to compute the asymptotic of the number of primes x ≤ p ≤ x + xε with Frobenius conjugacy class in E equal to C. The level of difficulty grows as ε becomes smaller. Assuming the Generalized Riemann Hypothesis, one can merely reach the regime 1 ≥ ε > 1/2. We establish a function field analogue of the Chebotarev theorem in short intervals for any ε > 0. Our result is valid in the limit when the size of the finite field tends to ∞ and when the extension is tamely ramified at infinity. The methods are based on a higher dimensional explicit Chebotarev theorem and applied in a much more general setting of arithmetic functions, which we name G-factorization arithmetic functions. [ABSTRACT FROM AUTHOR]

Published
2020
Full Text
View/download PDF

91. Prime tuples in function fields

Author

Bary-Soroker, Lior

Subjects
Algebra and Number Theory
Abstract

How many prime numbers are there? How are they distributed among other numbers? These are questions that have intrigued mathematicians since ancient times. However, many questions in this area have remained unsolved, and seemingly unsolvable in the forseeable future. In this snapshot, we will discuss one such problem, the Twin Prime Conjecture, and a quantitative version of it known as the Hardy–Littlewood Conjecture. We will also see that these and other questions about prime numbers can be extended to questions about function fields, and discuss recent progress which has been made to answer them in this context.

Published
2016
Full Text
View/download PDF

92. On the remainder in Taylor theorem

Author

Alfaro, Ricardo, Lixing Han, Schilling, Kenneth, and Bary-Soroker, Lior

Subjects
College students -- Education, Polynomials -- Analysis, Series, Taylor's -- Tests, problems and exercises, Theorems (Mathematics) -- Analysis, Education, Mathematics
Published
2009

96. Correlations of Sums of Two Squares and Other Arithmetic Functions in Function Fields.

Author

Bary-Soroker, Lior and Fehm, Arno

Subjects
*SUM of squares, *ARITHMETIC functions, *FINITE fields, *LOGICAL prediction
Abstract

We investigate a function field analogue of a recent conjecture on autocorrelations of sums of two squares by Freiberg, Kurlberg, and Rosenzweig, which generalizes an older conjecture by Connors and Keating. In particular, we provide extensive numerical evidence and prove it in the large finite field limit. Our method can also handle correlations of other arithmetic functions and we give applications to (function field analogues of) the average of sums of two squares on shifted primes, and to autocorrelations of higher divisor functions twisted by a quadratic character. [ABSTRACT FROM AUTHOR]

Published
2019
Full Text
View/download PDF

97. Hilbertian fields and Galois representations

Author

Bary-Soroker, Lior, Fehm, Arno, Wiese, Gabor, Bary-Soroker, Lior, Fehm, Arno, and Wiese, Gabor

Abstract

We prove a new Hilbertianity criterion for fields in towers whose steps are Galois with Galois group either abelian or a product of finite simple groups. We then apply this criterion to fields arising from Galois representations. In particular we settle a conjecture of Jarden on abelian varieties.

Published
2016
Full Text
View/download PDF

100. PRIME POLYNOMIALS IN SHORT INTERVALS AND IN ARITHMETIC PROGRESSIONS

Author

Bank, Efrat, Bary-Soroker, Lior, Rosenzweig, Lior, Bank, Efrat, Bary-Soroker, Lior, and Rosenzweig, Lior

Abstract

In this paper we establish function field versions of two classical conjectures on prime numbers. The first says that the number of primes in intervals (x,x x(is an element of)] is about x(is an element of)/logx. The second says that the number of primes p < x in the arithmetic progression p equivalent to a (mod d), for d < x(1-delta), is about pi(x)/phi(d), where phi is the Euler totient function. More precisely, for short intervals we prove: Let k be a fixed integer Then pi(q)(I(f,is an element of)) similar to #I(f,is an element of)/k, q -> infinity holds uniformly for all prime powers q, degree k monic polynomials is an element of F-a[t] and is an element of(0)(f, q) <= is an element of, where is an element of(0) is either 1/k, or 2/k if p vertical bar k(k-1), or 3/k if further p = 2 and deg f ' <= 1. Here I (f, is an element of) = {g is an element of F-q[t] vertical bar deg( f - g) <= is an element of deg f }, and pi(q)(I(f, is an element of)) denotes the number of prime polynomials in I (f, 6). We show that this estimation fails in the neglected cases. For arithmetic progressions we prove: let k be a fixed integer. Then pi(q)(k; D,f) similar to pi(q)(k)/phi(D), q -> infinity, holds uniformly for all relatively prime polynomials D, f is an element of F-q[t] satisfying parallel to D parallel to <= q(k)(1-(delta 0)) where delta(0) is either 3/k or 4/k if p = 2 and (f /D)' is a constant. Here pi(q)(k) is the number of degree k prime polynomials and pi(a)(k; D, f) is the number of such polynomials in the arithmetic progression P equivalent to f (mod D). We also generalize these results to arbitrary factorization types., QC 20150325

Published
2015
Full Text
View/download PDF

Catalog

Books, media, physical & digital resources
See catalog results