Now showing items 1-7 of 7

• #### Rational points and lines on cubic hypersurfaces ﻿

(2023-10-02)
We study the solubility of cubic diophantine equations. In the first chapter, we discuss the convergence of the singular series of a cubic form, which is a central object in the study of the solutions of the associated ...
• #### Two Cases of Artin's Conjecture ﻿

(2021-02-25)
Let $f_1, \dots, f_R$ be forms of degree $k_1, \dots, k_R$ in $s$ variables. A generalised version of a conjecture by Artin states that the equations $f_1= \dots=f_R=0$ have a non-trivial $p$-adic solution for all primes ...
• #### The Barban-Davenport-Halberstam for tuples of k-free numbers ﻿

(2020-10-23)
An asymptotic formula for variance of tuples of k-free numbers in arithmetic progressions
• #### The distribution of rational points on some projective varieties ﻿

(2020-01-09)
This thesis is concerned with establishing Manin's conjecture on the distribution of ratinal points for a certain class of bihomogeneous varieties. It generalizes work of Vaughan on the representation of integers as sum ...
• #### Diophantine Representation in Thin Sequences ﻿

(2016-08-24)
In this work we investigate conditions under which forms of arbitrary degree represent almost all elements of thin sequences (especially the set of squares). Stronger results are given for forms of degree 3 and 4.
• #### Quadratische Diophantische Gleichungen über algebraischen Zahlkörpern ﻿

(2015-04-23)
A search bound for the smallest solution of a quadratic diophantine equation over number fields in at least three variables is established.
• #### Diophantine Equations in Many Variables ﻿

(2014-11-06)
Let K denote a p-adic field and $F_1,..,F_r \in k[x_1, . . . , x_n]$ be forms with respective degrees $d_1, . . . , d_r$. A contemporary version of a conjecture attributed to E. Artin states that $F_1, . . . , F_r$ have a ...