On the irreducibility of sums of rational functions with separated variables (Q1320031)

Let \(k\) be a field of characteristic \(p\geq 0\), let \(n\geq 3\) be a natural number. Let \(f_ i= F_ i/H_ i\) where \(\text{gcd} (F_ i,H_ i)=1\). The problem is whether the numerator of the rational function \(f(X_ 1,\dots, X_ n) = \sum_{i=1}^ n f_ i(X_ i)\) is an irreducible polynomial. The answer is positive with the exception of a special case. Let \(p>0\), \(L\in k[x]\) be an additive polynomial of degree \(>1\), i.e. \(L(X+Y)= L(X)+ L(Y)\), and \(\widetilde{f}_ i\in k(x)\) be such that \(f_ i(x)= L(\widetilde{f}_ i(x))\). In this case the numerator of \(f\) is reducible. This result was first proved in [\textit{A. Schinzel}, Pac. J. Math. 118, 531-563 (1985; Zbl 0571.12011)]. The author also develops some ideas of \textit{M. Fried} [J. Pure Appl. Algebra 48, 9-22 (1987; Zbl 0646.13006)] to obtain a new proof to all characteristics. The paper contains a detailed historical review.