$Q$-difference analogue of the Stothers-Mason theorem

Abstract: In this paper, we give a new definition of the $q$-weight of zeros, which reduces to the multiplicity of zeros as $q\to 1$. Furthermore, we obtain a $q$-difference version of the Stothers-Mason theorem by means of the new definition of the $q$-difference radical, which covers the classical Stothers-Mason theorem as $q\to 1$. As applications, we study the polynomial solutions of $q$-difference Fermat type functional equations.