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A Note on Hilbert's Tenth Problem

Published online by Cambridge University Press:  20 November 2018

Z. A. Melzak*
Affiliation:
McGill University
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The tenth problem on Hilbert's well known list [1] is the following.

(H 10) For an arbitrary polynomial P = P(x1,x2,…,xn) with integer coefficients to determine whether or not the equation P = 0 has a solution in integers.

By 'integers' we always mean 'rational integers'. The problem (H 10) is still unsolved but it appears likely that no decision procedure exists; in this connection see [2]. It will be shown here that (H 10) is equivalent to deciding whether or not every member of a certain given countable sec of rational functions of a single variable x is absolutely monotonie. We recall that f(x) is absolutely monotonie in I if f(x) possesses non-negative derivatives of all orders at every x ∊ I.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1960

References

1. Hilbert, D., Mathematical problems, Bull. Amer. Math, Soc. 8 (1901), 437-479.Google Scholar
2. Davis, M., Computability and Unsolvability, (New York, 1958).Google Scholar