# Quadratic and cubic diophantine inequalities.

Freeman, Donald Eric

1999

## Abstract

In this thesis, we consider conditions under which certain quadratic and cubic Diophantine inequalities have solutions. Schmidt has proved that if <italic>d</italic><sub>1</sub>, ..., <italic>d<sub>r</sub></italic> are odd positive integers, then there is an integer f (<italic>d</italic><sub>1</sub>, ..., <italic>d<sub>r</sub></italic>) depending only on <italic>d</italic><sub>1</sub>, ..., <italic>d<sub> r</sub></italic> so that if <italic>s</italic> is any integer with <italic> s</italic> ≥ f (<italic>d</italic><sub>1</sub>, ..., <italic>d<sub>r</sub></italic>) and <italic>F</italic><sub>1</sub>, ..., <italic>F<sub>r</sub></italic> are real forms in <italic>s</italic> variables of degrees <italic>d</italic><sub> 1</sub>, ..., <italic>d<sub>r</sub></italic>, respectively, then there is a nonzero integral solution <bold>x</bold> of the simultaneous Diophantine inequalities ∣F1<b>x</b></fen> ∣<1,&ldots;,∣FR<b>x</b></fen> ∣<1 . We give explicit bounds for f if all of the forms are cubics, and we prove a theorem for quadratic forms of a flavor similar to Schmidt's result. We show that if s≥100R</fen> 100R</fen>200R , then given any <italic>R</italic> real cubic forms <italic>C</italic><sub> 1</sub>, ..., <italic>C<sub>R</sub></italic> in <italic> s</italic> variables, there is a nonzero integral solution <bold>x</bold> of the simultaneous Diophantine inequalities ∣C1<b>x</b></fen> ∣<1,&ldots;,∣CR<b>x</b></fen> ∣<1 . The case <italic>R</italic> = 1 is treated with particular care, and it is shown that one cubic Diophantine inequality in at least 358,523,548 variables has a non-trivial integral solution, improving the previously known bounds significantly. Concerning quadratic inequalities, we consider real positive definite quadratic forms <italic>Q</italic>(<bold>x</bold>) with algebraic coefficients. Suppose that the ratios of the coefficients of <italic>Q</italic>(<bold>x </bold>) are not all rational. It is shown that there is a number <italic> s</italic><sub>0</sub> such that if <italic>s</italic> > <italic>s</italic><sub> 0</sub>, then for any positive e there is a large number <italic>M</italic><sub>0</sub>( e ) such that for all <italic>M</italic> ≥ <italic>M</italic><sub>0</sub>( e ), there exists an integral vector <bold>x</bold> with ∣Q<b>x</b></fen>-M∣<e . In fact we may take <italic>s</italic><sub>0</sub> = 416. We note that the author has recently discovered that a stronger result than this has been claimed by other workers. The techniques used to prove the above results include the Hardy-Littlewood and Davenport-Heilbronn methods, and efficient diagonalization processes.## Subjects

Cubic Forms Diophantine Inequalities Quadratic Forms

## Types

Thesis

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