dc.contributor.author Xiao, Stanley Yao dc.date.accessioned 2016-08-17 19:30:15 (GMT) dc.date.available 2016-08-17 19:30:15 (GMT) dc.date.issued 2016-08-17 dc.date.submitted 2016-08-15 dc.identifier.uri http://hdl.handle.net/10012/10649 dc.description.abstract In this thesis we study several problems related to the representation of integers by binary forms and counting rational points on algebraic varieties. In particular, we establish an asymptotic formula for $R_F(Z)$, the number of integers of absolute value up to $Z$ which can be represented by a binary form $F$ with integer coefficients, degree $d \geq 3$, and non-zero discriminant. We give superior results when $d = 3$ or $4$, which completely resolves the cases considered by Hooley. We establish an asymptotic formula for the number of pairs $(x,y) \in \bZ^2$ such that $F(x,y)$ is $k$-free, whenever $F$ satisfies certain necessary conditions and $k > 7d/18$. Finally, we give various results on the arithmetic of certain cubic and quartic surfaces as well as general methods to estimate the number of rational points of bounded height on algebraic varieties. In particular, we give a bound for the density of rational points on del Pezzo surfaces of degree $2$. These results depend on generalizations of Salberger's global determinant method in various settings. en dc.language.iso en en dc.publisher University of Waterloo en dc.subject Binary forms en dc.subject determinant method en dc.subject power-free values en dc.subject representation of integers by binary forms en dc.title Some results on binary forms and counting rational points on algebraic varieties en dc.type Doctoral Thesis en dc.pending false uws-etd.degree.department Pure Mathematics en uws-etd.degree.discipline Pure Mathematics en uws-etd.degree.grantor University of Waterloo en uws-etd.degree Doctor of Philosophy en uws.contributor.advisor Stewart, Cameron uws.contributor.affiliation1 Faculty of Mathematics en uws.published.city Waterloo en uws.published.country Canada en uws.published.province Ontario en uws.typeOfResource Text en uws.peerReviewStatus Unreviewed en uws.scholarLevel Graduate en
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