# Quadratic and cubic diophantine inequalities.

 dc.contributor.author Freeman, Donald Eric dc.contributor.advisor Wooley, Trevor dc.date.accessioned 2016-08-30T17:49:57Z dc.date.available 2016-08-30T17:49:57Z dc.date.issued 1999 dc.identifier.uri http://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.uri https://hdl.handle.net/2027.42/131649 dc.description.abstract In this thesis, we consider conditions under which certain quadratic and cubic Diophantine inequalities have solutions. Schmidt has proved that if d1, ..., dr are odd positive integers, then there is an integer f (d1, ..., dr) depending only on d1, ..., d r so that if s is any integer with s ≥ f (d1, ..., dr) and F1, ..., Fr are real forms in s variables of degrees d 1, ..., dr, respectively, then there is a nonzero integral solution x of the simultaneous Diophantine inequalities ∣F1x ∣<1,&ldots;,∣FRx ∣<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 100R200R , then given any R real cubic forms C 1, ..., CR in s variables, there is a nonzero integral solution x of the simultaneous Diophantine inequalities ∣C1x ∣<1,&ldots;,∣CRx ∣<1 . The case R = 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 Q(x) with algebraic coefficients. Suppose that the ratios of the coefficients of Q(x ) are not all rational. It is shown that there is a number s0 such that if s > s 0, then for any positive e there is a large number M0( e ) such that for all MM0( e ), there exists an integral vector x with ∣Qx-M∣s0 = 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.extent 194 p. dc.language English dc.language.iso EN dc.subject Cubic Forms dc.subject Diophantine dc.subject Inequalities dc.subject Quadratic Forms dc.title Quadratic and cubic diophantine inequalities. dc.type Thesis dc.description.thesisdegreename Ph.D. dc.description.thesisdegreediscipline Mathematics dc.description.thesisdegreediscipline Pure Sciences dc.description.thesisdegreegrantor University of Michigan, Horace H. Rackham School of Graduate Studies dc.description.bitstreamurl http://deepblue.lib.umich.edu/bitstream/2027.42/131649/2/9929825.pdf dc.owningcollname Dissertations and Theses (Ph.D. and Master's)
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