Diophantine Equations in Many Variables

by Jan Henrik Dumke
Doctoral thesis
Date of Examination:2014-10-08
Date of issue:2014-11-06
Advisor:Prof. Dr. Jörg Brüdern
Referee:Prof. Dr. Jörg Brüdern
Referee:Prof. Dr. Valentin Blomer
Persistent Address: http://dx.doi.org/10.53846/goediss-4773

 

 

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Abstract

English

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 common non-trivial zero whenever $n > d_1^2 + · · · + d_r^2$. We prove this for a single quintic form $(i.e.~ r = 1, d_1 = 5)$, provided that the cardinality of the residue class field exceeds 9. We also verify the conjecture for a system comprising a cubic and a quadratic form $(i.e.~r = 2, d_1 = 3, d_2 = 2)$, whenever the residue class field is of characteristic at least 13 and has more than 37 elements.
Keywords: p-adic forms; forms in many variables
 
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