A note on Gao's algorithm for polynomial factorization (Q633632)

For a bivariate polynomial \(f(X,Y)\) in \(\mathbb F[X,Y]\) of bidegree \((m,n)\), with \(\mathbb F\) field of characteristic \(0\) or \(>(2m-1)n\), the well-known factorization algorithm by \textit{S. Gao} [Math. Comput. 72, No. 242, 801--822 (2003; Zbl 1052.12006)] studies the property of a special vector space. More precisely, \(G\) is the vector space of the solutions of a differential equation defined from \(f(X,Y)\). The dimension of \(G\) is equal to \(r\), the number of absolute factors of \(f(X,Y)\); Gao's algorithm also provides a basis for this vector space. In this paper, the authors remove the hypothesis on the caracteristic of \(\mathbb F\), defining a subspace of \(G\) of dimension \(r\) in which Gao's construction holds. Furthermore, the authors indentify a second subspace of \(G\) leading to an analogous theory for the rational factorization of \(f(X,Y)\).