A note on Gao's algorithm for polynomial factorization

Authors:
Carlos Hoppen;Virginia M. Rodrigues;Vilmar Trevisan
Affiliations:
Instituto de Matemática, Universidade Federal do Rio Grande do SulAvenida Bento Gonçalves, 9500, 91509-900 Porto AlegreRS, Brazil;Departamento de Matemática, Pontifícia Universidade CatólicaAvenida Ipiranga, 6681, 91530-000 Porto AlegreRS, Brazil;Instituto de Matemática, Universidade Federal do Rio Grande do SulAvenida Bento Gonçalves, 9500, 91509-900 Porto AlegreRS, Brazil
Venue:
Theoretical Computer Science
Year:
2011

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Abstract

Shuhong Gao (2003) [6] has proposed an efficient algorithm to factor a bivariate polynomial f over a field F. This algorithm is based on a simple partial differential equation and depends on a crucial fact: the dimension of the polynomial solution space G associated with this differential equation is equal to the number r of absolutely irreducible factors of f. However, this holds only when the characteristic of F is either zero or sufficiently large in terms of the degree of f. In this paper we characterize a vector subspace of G for which the dimension is r, regardless of the characteristic of F, and the properties of Gao's construction hold. Moreover, we identify a second vector subspace of G that leads to an analogous theory for the rational factorization of f.