On polynomial congruences

Abstract

Abstract. We deal with functions which fulfil the condition Δ n+1 h ϕ(x) ∈ Z for all x, h taken from some linear space V . We derive necessary and sufficient conditions for such a function to be decent in the following sense: there exist functions f :Mathematics Subject Classification. Primary 39B99; Secondary 39A70.

On linear functional equations modulo $${\mathbb {Z}}$$

“…Finally, we formulate and prove two corollaries of our theorems as well as some results of Lewicka [20] on solutions of linear congruences satisfying certain regularity properties. ϕ 0 , .…”

“…which she called polynomial congruence of degree n (cf. [20]). Analogously to the case connected to the Cauchy equation, a function ϕ : V → R is said to be a decent solution of the polynomial congruence of degree n if it satisfies ( 9) and there exist functions f : V → R and g : V → Z, such that f is a polynomial function of degree n and ϕ = f + g [20].…”

“…Our investigations were mainly motivated by results of J. A. Baker, K. Baron, J. Brzdȩk, M. Sablik, P. Volkmann published in the papers [1,3,4,8], respectively, connected to the Cauchy equation modulo Z, as well as, by studies of A. Lewicka on polynomial functional equations modulo Z [20].…”