Characterizations and constructions of n-to-1 mappings over finite fields

n-to-1 mappings have wide applications in many areas, especially in cryptography, finite geometry, coding theory and combinatorial design. In this paper, many classes of n-to-1 mappings over finite fields are studied. First, we provide a characterization of general n-to-1 mappings over ${\mathbb{F}}_{p^m}$ by means of the Walsh transform. Then, we completely determine 3-to-1 polynomials with degree no more than 4 over ${\mathbb{F}}_{p^m}$. Furthermore, we obtain an AGW-like criterion for characterizing some close relationship between the n-to-1 property of a mapping over finite set A and that of another mapping over a subset of A. Finally, we apply the AGW-like criterion into several forms of polynomials and obtain some explicit n-to-1 mappings. Especially, three explicit constructions of the form $x^{r}h\left(x^s\right)$ from the cyclotomic perspective, and several classes of n-to-1 mappings of the form $g(x^{q^k}-x+\delta )+cx$ are provided.