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Quadratic and cubic diophantine inequalities.

dc.contributor.authorFreeman, Donald Eric
dc.contributor.advisorWooley, Trevor
dc.date.accessioned2016-08-30T17:49:57Z
dc.date.available2016-08-30T17:49:57Z
dc.date.issued1999
dc.identifier.urihttp://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:9929825
dc.identifier.urihttps://hdl.handle.net/2027.42/131649
dc.description.abstractIn 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> &ge; 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 &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><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 &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><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> &ge; <italic>M</italic><sub>0</sub>( 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><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.
dc.format.extent194 p.
dc.languageEnglish
dc.language.isoEN
dc.subjectCubic Forms
dc.subjectDiophantine
dc.subjectInequalities
dc.subjectQuadratic Forms
dc.titleQuadratic and cubic diophantine inequalities.
dc.typeThesis
dc.description.thesisdegreenamePh.D.
dc.description.thesisdegreedisciplineMathematics
dc.description.thesisdegreedisciplinePure Sciences
dc.description.thesisdegreegrantorUniversity of Michigan, Horace H. Rackham School of Graduate Studies
dc.description.bitstreamurlhttp://deepblue.lib.umich.edu/bitstream/2027.42/131649/2/9929825.pdf
dc.owningcollnameDissertations and Theses (Ph.D. and Master's)


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