REDUCIBLE FIBERS OF POLYNOMIALS ARNAUD BODIN Abstract. We give an algorithm to find reducible fibers of a bi- variate polynomial. 1. Ruppert's equation, Gao's theorem Let F be any field of characteristic p. Let f ? F[x, y]. The aim of this note is to give a method to find the values c for which f ? c is reducible based on results of W. Ruppert and S. Gao, [Ru], [Ga]. We consider the following equation : (1) ∂ ∂y ( g f ? c ) = ∂ ∂x ( h f ? c ) , where g, h ? F[x, y] are unknows and f ? F[x, y], c ? F are given data. This equation can be “linearized” to the equivalent equation: (2) (f ? c) · ( ∂g ∂y ? ∂h ∂x ) = g · ∂f ∂y ? h · ∂f ∂x . Notice that this equation is linear in the coefficient of g and h. The bidegree of a polynomial g is deg g = (degx g, degy g). Let (m,n) be the bidgree of f . We impose the following restrictions: (3) deg g 6 (m ? 1, n), deg h 6 (m,n ? 1).
- stein fac- torization
- reducible fibers
- brianc¸on polynomial
- finding reducible
- pq ?
- polynomials via partial differential equations
- equation