Análogos del décimo problema de Hilbert = Analogues of hilbert's tenth problem.
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Date
2010
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Journal ISSN
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Publisher
Universidad de Concepción.
Abstract
The tenth problem in D. Hilbert's famous list asked the following :
Devise an algorithm to decide whether a polynomial equation
with integer coe cients has an integer solution.
(These equations are called Diophantine equations.) In the year 1970, 70
years after Hilbert posed it, Y. Matiyasevich (based on work of M. Davis,
J. Robinson and H. Putnam) proved that such an algorithm does not exist
[10].
Knowing that the decision problem for integer solutions of Diophantine
equations had a negative answer, the problem shifted to smaller classes of
equations. For example, it follows from Matiyasevich's negative answer that
there exists no algorithm to decide whether a system of second-degree Diophantine
equations has integral solutions; while, on the other hand, a result
of M. Presburger (1929) implies that an analogous algorithm for systems of
linear Diophantine equations exists [15].
Consider all systems of second degree Diophantine equations in where
every unknown appears squared in all of its ocurrences. These systems form
a subset of all systems of second degree Diophantine equations, and it can be
shown that every linear Diophantine equation can be written as such (just
because any integer can be written as x2+y2z2 for some integers x, y and
z). Thus, the decision problem for integer solutions to this kind of systems
of equations is \in between" the two already known results mentioned in the
previous paragraph. This problem is known as the Problem of representation
by diagonal quadratic forms, and is currently open.
Description
Tesis para optar al grado de Magíster en Matemática.
Keywords
Espacios de Hilbert, Problemas Ejercicios Etc., Ecuaciones Diofántinas, Algoritmos, Problemas Ejercicios Etc.