Complete characterization of a class of permutation trinomials in characteristic five

In this paper, we address an open problem posed by Bai and Xia in [2]. We study polynomials of the form f(x)=x^4q+1+λ _1x^5q+λ _2x^q+4 over the finite field 𝔽_5^k , which are not quasi-multiplicative equivalent to any of the known permutation polynomials in the literature. We find necessary and sufficient conditions on λ _1, λ _2 ∈𝔽_5^k so that f(x) is a permutation monomial, binomial, or trinomial of 𝔽_5^2k .