A new bound for an extension of Mason's theorem for functions of several variables
Using the generalized Wronskian, the authors obtain a new sharp bound for the generalized Mason's theorem [ \textit{H. N. Shapiro} and \textit{G. H. Sparer}, Commun. Pure. Appl. Math. 47, 711--718 (1994; Zbl 0818.11019)] for functions of several variables as follows (in the notation of the cited article), \[ \max_{1\leq i\leq n} \deg f_ i \leq (n-2)\biggl(\prod N_{i=1}^n(f_i)\biggr)- \tfrac{n-1}{2}\text{ if }k=0, \] \[ \max_{1\leq i\leq n} \deg f_ i \leq (n-k-1)\biggl(\prod N_{i=1}^n(f_i)\biggr)- \tfrac{n-k}{2}\text{ if }1\leq k\leq n-2, \] where \(k\) out of the \(n\)-polynomials are constant. They also show that the Diophantine equation (the generalized Fermat-Catalan equation) \[ a_{1}^{m_1} + a_{2}^{m_2} + \cdots + a_{n-1}^{m_{n-1}} = a_{n}^{m_{n}}, \] where \( a_{1}, a_{2}, \ldots, a_{n} \in \mathbb C[x_{1}, \ldots, x_{l}] \) , such that \(k\) out of the \(n\)-polynomials \( (k \leq n-2) \) are constant, and \( m_{1}, m_{2}, \ldots, m_{n} \in \mathbb N, \) under certain conditions for \( a_{i}(i = 1, \ldots, n) \), has no non-constant solution.
scientific article; zbMATH DE number 2119260