Event № 281
How many numbers between X and X+H are square-free, where X is large and H > X^ε? In how many ways can a large number N be given as a sum N = x^k + r of a positive k-th power and a positive square-free number r? In full generality, both questions are still mostly open. They can be seen as special cases of a more general question - how many values of a polynomial f(x) are square-free, where the coefficients of the polynomial are much larger than the values the argument assumes? We answer these questions in the function field setting, over a fixed finite field with degrees going to infinity, following the techniques of Poonen and Lando, who solved similar questions for polynomials with fixed coefficients.