Waring’s problem with digital restrictions in $$ \mathbb{F} $$ q[X]

Abstract

ABSTRACT. We present a generalization of a result due to Thuswaldner and Tichy to the ring of polynomials over a finite fields. In particular, we want to show that every polynomial of sufficiently large degree can be represented as sum of kth powers, where the bases evaluated on additive functions meet certain congruence restrictions.

Additive functions for number systems in function fields

“…Analogs of the two distribution theorems above were shown for this setting by Drmota and Gutenbrunner [6]. Waring's Problem with this digitally restricted set was solved by the first author [12] where the Weyl sum estimates came from the two authors of the present paper [13].…”

“…The proofs of our results are based on the proofs of the corresponding results for number systems in F q [X] in the sense of Kovács and Pethő [10]. In particular, the proofs of Theorem 2.3 and Theorem 2.4 will follow Drmota and Gutenbrunner [6], the proof of Theorem 2.5 will follow Madritsch and Thuswaldner [13], and the proof of Theorem 2.6 will follow Madritsch [12]. New difficulties occur in our more general setting.…”