On the power-free values of polynomials in two variables: II |
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Authors: | C Hooley |
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Institution: | Cardiff School of Mathematics, Cardiff University, Senghennydd Road, Cardiff, CF24 4AG, United Kingdom |
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Abstract: | The following theorem is proved. Suppose that for integers r,s?2, f(x,y) is an inhomogeneous polynomial of degree r with rational integral coefficients that is irreducible over the rationals, that is not a polynomial in a linear combination of x and y, that has no fixed sth power divisors other than 1, and that is a product of linear factors over some extension field of the rationals. Then, if N(X) denote the number of integers m,n of magnitude not exceeding X for which f(m,n) is sth power-free, the asymptotic formula |
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