New results on n-to-1 mappings over finite fields

n-to-1 mappings have many applications in cryptography, finite geometry, coding theory and combinatorial design. In this paper, we first use cyclotomic cosets to construct several kinds of n-to-1 mappings over $F_q^{⁎}$. Then we characterize a new form of AGW-like criterion, and use it to present many classes of n-to-1 polynomials with the form $x^{r}h\left(x^{q-1}\right)$ over $F_{q^2}^{⁎}$. Finally, by using monomials on the cosets of a subgroup of ${\mu }_{q+1}$ and another form of AGW-like criterion, we show some n-to-1 trinomials over $F_{q^2}^{⁎}$.