Resumen de On the representation of k-free integers by binary forms

Cameron L. Stewart, Stanley Yao Xiao

• Let F be a binary form with integer coefficients, non-zero discriminant and degree d with d at least 3 and let r denote the largest degree of an irreducible factor of F over the rationals. Let k be an integer with k≥2 and suppose that there is no prime p such that pk divides F(a,b) for all pairs of integers (a,b). Let RF,k(Z) denote the number of k-free integers of absolute value at most Z which are represented by F. We prove that there is a positive number CF,k such that RF,k(Z) is asymptotic to CF,kZ2/d provided that k exceeds 7r/18 or (k,r) is (2,6) or (3,8).