Squarefree values of multivariable polynomials

Squarefree Values of Multivariable Polynomials

Given f ∈ Z[x1, . . . , xn], we compute the density of x ∈ Z such that f(x) is squarefree, assuming the abc conjecture. Given f, g ∈ Z[x1, . . . , xn], we compute unconditionally the density of x ∈ Z such that gcd(f(x), g(x)) = 1. Function field analogues of both results are proved unconditionally. Finally, assuming the abc conjecture, given f ∈ Z[x], we estimate the size of the image of f({1, ...

Squarefree Values of Polynomials

where m is any integer > n:We will be interested in estimating the number of polynomials f(x) for which there exists an integer m such that f(m) is squarefree. This property should hold for all polynomials f(x) for which Nf is squarefree. However, this seems to be very di cult to establish. Nagel [8] showed that if f(x) 2 Z[x] is an irreducible quadratic and Nf is squarefree, then f(m) is squar...

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Squarefree Values of Polynomials and the Abc-conjecture

Given > 0, there is a number C() such that for any relatively prime positive integers a, b, c with a + b = c, we have c < C() ? Q(abc) 1+ ; (1:1) where Q(k) = Q pjk p denotes the product of the distinct primes dividing k. See e.g. 11] for a discussion of the history and implications of the conjecture. It can of course be expressed in terms of not necessarily positive integers a, b, c (which may...

A construction of polynomials with squarefree discriminants

For any integer n ≥ 2 and any nonnegative integers r, s with r + 2s = n, we give an unconditional construction of infinitely many monic irreducible polynomials of degree n with integer coefficients having squarefree discriminant and exactly r real roots. These give rise to number fields of degree n, signature (r, s), Galois group Sn, and squarefree discriminant; we may also force the discrimina...