Quadratic and cubic diophantine inequalities.

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>₁, ..., <italic>d_r</italic> are odd positive integers, then there is an integer f (<italic>d</italic>₁, ..., <italic>d_r</italic>) depending only on <italic>d</italic>₁, ..., <italic>d_r</italic> so that if <italic>s</italic> is any integer with <italic> s</italic> &ge; f (<italic>d</italic>₁, ..., <italic>d_r</italic>) and <italic>F</italic>₁, ..., <italic>F_r</italic> are real forms in <italic>s</italic> variables of degrees <italic>d</italic>₁, ..., <italic>d_r</italic>, respectively, then there is a nonzero integral solution <bold>x</bold> of the simultaneous Diophantine inequalities &mid;F1<b>x</b></fen> &mid;<1,&ldots;,&mid;FR<b>x</b></fen> &mid;<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&ge;100R</fen> 100R</fen>200R , then given any <italic>R</italic> real cubic forms <italic>C</italic>₁, ..., <italic>C_R</italic> in <italic> s</italic> variables, there is a nonzero integral solution <bold>x</bold> of the simultaneous Diophantine inequalities &mid;C1<b>x</b></fen> &mid;<1,&ldots;,&mid;CR<b>x</b></fen> &mid;<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>₀ such that if <italic>s</italic> > <italic>s</italic>₀, then for any positive e there is a large number <italic>M</italic>₀( e ) such that for all <italic>M</italic> &ge; <italic>M</italic>₀( e ), there exists an integral vector <bold>x</bold> with &mid;Q<b>x</b></fen>-M&mid;<e . In fact we may take <italic>s</italic>₀ = 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.