Regis de la Breteche
(Universite Paris Diderot, Institut de Mathematiques de Jussieu-Paris Rive Gauche)

Largest prime factor of integer value of polynomial of degree 4

Let $P^+(n)$ denote the largest prime factor of the integer n. Using Heath-Brown and Dartyge methods, we prove that for all even unitary irreducible quartic polynomials ^H $F$ with integral coefficients and an associated Galois group isomorphic to $V_4$, there exists a positive constant c >0 such that the set of integers $n\leq x$ satisfying $$P^+( F(n) )\leq x^{1+c_F}$$ has a positive density. Such a result was recently proved by Dartyge for $F(n)=n^4-n^2+1$.