On polynomial congruences (Q5895085)

The main concern of the author is studying functions \(\varphi: V\rightarrow \mathbb{R}\) which are fulfilling a less restrictive condition than \[ \Delta_h^{n+1}f(x) =0 \quad x, h \in V, \] which are called polynomial functions of degree \(n\). Using the definition of polynomial congruence of degree \(n\) which is \(\Delta_h^{n+1}\varphi(x) \in \mathbb{Z}\), the following in the literature so-called Cauchy's congruence (or Cauchy equation modulo \(\mathbb{Z}\)), namely \[ \varphi(x+y)-\varphi(x)-\varphi(y) \in \mathbb{Z} , \quad x, y \in V , \quad \varphi: V\rightarrow \mathbb{R}, \] is considered, and the problem of decency in the sense of Baker solutions is studied.