Hilbert's Tenth Problem

Couverture
MIT Press, 1993 - 264 pages
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At the 1900 International Congress of Mathematicians, held that year in Paris, theGerman mathematician David Hilbert put forth a list of 23 unsolved problems that he saw as being thegreatest challenges for twentieth-century mathematics. Hilbert's 10th problem, to find a method(what we now call an algorithm) for deciding whether a Diophantine equation has an integralsolution, was solved by Yuri Matiyasevich in 1970. Proving the undecidability of Hilbert's 10thproblem is clearly one of the great mathematical results of the century.This book presents the full,self-contained negative solution of Hilbert's 10th problem. In addition it contains a number ofdiverse, often striking applications of the technique developed for that solution (scatteredpreviously in journals), describes the many improvements and modifications of the original proof -since the problem was "unsolved" 20 years ago, and adds several new, previously unpublishedproofs.Included are numerous exercises that range in difficulty from the elementary to smallresearch problems, open questions,and unsolved problems. Each chapter concludes with a commentaryproviding a historical view of its contents. And an extensive bibliography contains references toall of the main publications directed to the negative solution of Hilbert's 10th problem as well asthe majority of the publications dealing with applications of the solution.Intended for youngmathematicians, Hilbert's 10th Problem requires only a modest mathematical background. A few lesswell known number-theoretical results are presented in the appendixes. No knowledge of recursiontheory is presupposed. All necessary notions are introduced and defined in the book, making itsuitable for the first acquaintance with this fascinating subject.Yuri Matiyasevich is Head of theLaboratory of Mathematical Logic, Steklov Institute of Mathematics, Russian Academy of Sciences,Saint Petersburg.

 

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Table des matières

Diophantine Complexity
153
Decision Problems in Calculus
165
Other Applications of Diophantine Representations
181
Appendix
199
Hints to the Exercises
205
Droits d'auteur

Expressions et termes fréquents

Fréquemment cités

Page 101 - Occasionally it happens that we seek the solution under insufficient hypotheses or in an incorrect sense, and for this reason do not succeed. The problem then arises : to show the impossibility of the solution under the given hypotheses, or in the sense contemplated.
Page 102 - ... originally intended. It is probably this important fact along with other philosophical reasons that gives rise to the conviction (which every mathematician shares, but which no one has as yet supported by a proof) that every definite mathematical problem must necessarily be susceptible of an exact settlement, either in the form of an actual answer to the question asked, or by the proof of the impossibility of its solution and therewith the necessary failure of all attempts.
Page 102 - In later mathematics, the question as to the impossibility of certain solutions plays a preeminent part, and we perceive in this way that old and difficult problems, such as the proof of the axiom of parallels, the squaring of the circle, or the solution of equations of the fifth degree by radicals have finally found fully satisfactory and rigorous solutions, although in another sense [than that originally intended. It is probably this important fact along with other philosophical reasons that gives...
Page 102 - ... isosceles right triangle is irrational. In later mathematics, the question as to the impossibility of certain solutions plays a preeminent part, and we perceive in this way that old and difficult problems, such as the proof of the axiom of parallels, the squaring of the circle, or the solution of equations of the fifth degree by radicals have finally found fully satisfactory and rigorous solutions, although in another sense than that originally intended.
Page xix - ... 10. Determination of the Solvability of a Diophantine Equation Given a diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: To devise a process according to which it can be determined by a fmite number of operations whether the equation is solvable in rational integers. 11. Quadratic Forms with any Algebraic Numerical Coefficients Our present knowledge of the theory of quadratic number fieldsi...
Page 117 - It states that every even number greater than 2 is the sum of two primes in at least one way.
Page 69 - These results are superficially related to Hilbert's tenth problem on (ordinary, ie, non-exponential) Diophantine equations. The proof of the authors' results, though very elegant, does not use recondite facts in the theory of numbers nor in the theory of re [recursively enumerable] sets, and so it is likely that the present result is not closely connected with Hilberfs tenth problem.
Page xiv - ... definitions concerning which we were able to show that their being Diophantine would imply the same for all recursively enumerable sets. Hilary and I resolved to seek other opportunities to work together, and we were able to obtain support for our research during the three summers of 1958, 1959, and 1960. We had a wonderful time. We talked constantly about everything under the sun. Hilary gave me a quick course in classical European philosophy, and I gave him one in functional analysis. We talked...
Page xiii - I would emphasize the important consequences that would follow from either a proof or a disproof of the existence of such an equation. Inevitably during the question period I would be asked for my own opinion as to how matters would turn out, and I had my reply ready: "I think that Julia Robinson's hypothesis is true, and it will be proved by a clever young Russian.
Page 69 - The proof of the authors' result, though very elegant, does not use recondite facts in the theory of numbers nor in the theory of re [recursively enumerable] sets, and so it is likely that the present result is not closely connected with Hilbert's tenth Problem. Also it is not altogether plausible that all (ordinary) Diophantine problems are uniformly reducible to those in a fixed number of variables of fixed degree, which would be the case if all re sets were Diophantine.

À propos de l'auteur (1993)

Yuri Matiyasevich is Head of the Laboratory of Mathematical Logic, Steklov Institute of Mathematics, Russian Academy of Sciences, Saint Petersburg.

Informations bibliographiques