P-regular Sequences And Primitive Roots Over Functions Fields With Positive Characteristic

The present master thesis contains two relatively independent parts. In the frstpart, we study p-regular sequences, and give a criterion for the p-regularity of p-adicvaluation of the values at natural numbers arguments of an analytic function defnedon the ring of p-adic integers. Then we apply the criterion to study quadratic linearrecurrent sequences, and obtain a necessary and sufcient condition so that the p-adicvaluation of the values of a quadratic linear recurrent sequence is a p-regular sequence.In particular, we show that for Lucas sequences—a family of quadratic linear recurrentsequences, including the famous Fibonacci sequence as a special case, the p-adicvaluation of their values are p-regular. In the second part, we study primitive rootsover a function feld with positive characteristic whose class number is equal to1, andobtain an upper bound of the degree of the least primitive roots.