國立虎尾科技大學 |

Hilbert's tenth problem : = diophantine classes and extensions to global fields /

Record Type:	Language materials, printed : Monograph/item
Title/Author:	Hilbert's tenth problem :/ Alexandra Shlapentokh.
Reminder of title:	diophantine classes and extensions to global fields /
Author:	Shlapentokh, Alexandra,
Description:	1 online resource (xiii, 320 pages) :digital, PDF file(s). :
Notes:	Title from publisher's bibliographic system (viewed on 05 Oct 2015).
Subject:	Hilbert's tenth problem. -
Online resource:	https://doi.org/10.1017/CBO9780511542954
ISBN:	9780511542954 (ebook)

Hilbert's tenth problem : = diophantine classes and extensions to global fields /
Shlapentokh, Alexandra,

Hilbert's tenth problem :diophantine classes and extensions to global fields /Alexandra Shlapentokh. - 1 online resource (xiii, 320 pages) :digital, PDF file(s). - New mathematical monographs ;7. - New mathematical monographs ;31..

Title from publisher's bibliographic system (viewed on 05 Oct 2015).

Introduction -- Diophantine classes: definitions and basic facts -- Diophantine equivalence and Diophantine decidability -- Integrality at finitely many primes and divisibility of order at infinitely many primes -- Bound equations for number fields and their consequences -- Units of rings of W-integers of norm 1 -- Diophantine classes over number fields -- Diophantine undecidability of function fields -- Bounds for function fields -- Diophantine classes over function fields -- Mazur's conjectures and their consequences -- Results of Poonen -- Beyond global fields -- Recursion (computability) theory -- Number theory.

In the late sixties Matiyasevich, building on the work of Davis, Putnam and Robinson, showed that there was no algorithm to determine whether a polynomial equation in several variables and with integer coefficients has integer solutions. Hilbert gave finding such an algorithm as problem number ten on a list he presented at an international congress of mathematicians in 1900. Thus the problem, which has become known as Hilbert's Tenth Problem, was shown to be unsolvable. This book presents an account of results extending Hilbert's Tenth Problem to integrally closed subrings of global fields including, in the function field case, the fields themselves. While written from the point of view of Algebraic Number Theory, the book includes chapters on Mazur's conjectures on topology of rational points and Poonen's elliptic curve method for constructing a Diophatine model of rational integers over a 'very large' subring of the field of rational numbers.

ISBN: 9780511542954 (ebook)Subjects--Topical Terms:

1443363
Hilbert's tenth problem.

LC Class. No.: QA242 / .S53 2007

Dewey Class. No.: 512.7

Hilbert's tenth problem : = diophantine classes and extensions to global fields /
LDR:02635nam a2200301 i 4500 001 1125253
003 UkCbUP
005 20151005020622.0
006 m|||||o||d||||||||
007 cr||||||||||||
008 240926s2007||||enk o ||1 0|eng|d
020 $a 9780511542954 (ebook)
020 $z 9780521833608 (hardback)
035 $a CR9780511542954
040 $a UkCbUP $b eng $e rda $c UkCbUP
050 0 0 $a QA242 $b .S53 2007
082 0 4 $a 512.7 $2 22
100 1 $a Shlapentokh, Alexandra, $e author. $3 1443362
245 1 0 $a Hilbert's tenth problem : $b diophantine classes and extensions to global fields / $c Alexandra Shlapentokh.
264 1 $a Cambridge : $b Cambridge University Press, $c 2007.
300 $a 1 online resource (xiii, 320 pages) : $b digital, PDF file(s).
336 $a text $b txt $2 rdacontent
337 $a computer $b c $2 rdamedia
338 $a online resource $b cr $2 rdacarrier
490 1 $a New mathematical monographs ; $v 7
500 $a Title from publisher's bibliographic system (viewed on 05 Oct 2015).
505 0 $a Introduction -- Diophantine classes: definitions and basic facts -- Diophantine equivalence and Diophantine decidability -- Integrality at finitely many primes and divisibility of order at infinitely many primes -- Bound equations for number fields and their consequences -- Units of rings of W-integers of norm 1 -- Diophantine classes over number fields -- Diophantine undecidability of function fields -- Bounds for function fields -- Diophantine classes over function fields -- Mazur's conjectures and their consequences -- Results of Poonen -- Beyond global fields -- Recursion (computability) theory -- Number theory.
520 $a In the late sixties Matiyasevich, building on the work of Davis, Putnam and Robinson, showed that there was no algorithm to determine whether a polynomial equation in several variables and with integer coefficients has integer solutions. Hilbert gave finding such an algorithm as problem number ten on a list he presented at an international congress of mathematicians in 1900. Thus the problem, which has become known as Hilbert's Tenth Problem, was shown to be unsolvable. This book presents an account of results extending Hilbert's Tenth Problem to integrally closed subrings of global fields including, in the function field case, the fields themselves. While written from the point of view of Algebraic Number Theory, the book includes chapters on Mazur's conjectures on topology of rational points and Poonen's elliptic curve method for constructing a Diophatine model of rational integers over a 'very large' subring of the field of rational numbers.
650 0 $a Hilbert's tenth problem. $3 1443363
650 0 $a Algebraic number theory. $3 672442
650 0 $a Diophantine equations. $3 792270
776 0 8 $i Print version: $z 9780521833608
830 0 $a New mathematical monographs ; $v 31. $3 1238333
856 4 0 $u https://doi.org/10.1017/CBO9780511542954