Two generalized \(abc\) theorems for \(\mathbb F_q[t]\) (Q2017205)

The Mason-Stothers theorem on univariate polynomials over an arbitrary field \(F\) is a polynomial analogue to what is still known as the abc-conjecture. It relates the maximum of the degrees of polynomials \(a,b,c\) with \(a+b=c\) to the number of distinct zeros of \(abc\) in the algebraic closure of \(F\). Here two analogous theorems are proved for univariate polynomials over a finite field \({\mathbb F}_q\), but considering irreducible factors in \({\mathbb F}_q[t]\) rather than in \(\overline{{\mathbb F}}_q[t]\). The two theorems use Wronskians by hyperderivatives and Galois derivatives, respectively. A new Wronskian criterion involving shift operators is introduced. The author also lists all known Wronskian criteria for linear independence of univariate polynomials over a finite field \({\mathbb F}_q\), involving both types of derivatives, and proves their equivalence.