Diophantine problems over global fields and a conjecture of Artin over function fields

by Leonhard Hochfilzer
Doctoral thesis
Date of Examination:2023-05-23
Date of issue:2023-05-26
Advisor:Prof. Dr. Damaris Schindler
Referee:Prof. Dr. Damaris Schindler
Referee:Prof. Dr. Jörg Brüdern
Persistent Address: http://dx.doi.org/10.53846/goediss-9906

 

 

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Abstract

English

This thesis considers three diophantine problems over different global fields and Artin's primitive root conjecture over function fields. In the first chapter, which is a joint project with Christian Bernert, the main result is that every cubic form over an imaginary quadratic extension of the rational numbers in at least $14$ variables represents zero nontrivially. This further implies that a rational cubic hypersurface of dimension at least $31$ contains a rational projective line. The second chapter, which is a joint effort with Jakob Glas, deals with diagonal cubic forms over the function field of the projective line over a finite field. Near optimal upper bounds are established when the number of variables is six, and as a result we improve Waring's problem in this context. In the third chapter we study the distribution of solutions of bounded height to a system of bihomogeneous equations of bidegree $(1,1)$ and $(2,1)$. We show that the relevant counting function of non-singular systems of this type satisfies the expected asymptotic if the number of variables grows only linearly in terms of the number of equations, as opposed to quadratically, which is what previous authors showed. Finally, in chapter 4, in a joint effort with Ezra Waxman Artin's primitive root conjecture is proved in full for any field which is an extension of finite transcendence degree over a finite field.
Keywords: Number theory; Diophantine equations; Artin's primitive root conjecture; Circle method; Function fields; Waring's problem
 
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