Squarefree values of multivariable polynomials
نویسندگان
چکیده
منابع مشابه
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, ...
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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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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...
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ژورنال
عنوان ژورنال: Duke Mathematical Journal
سال: 2003
ISSN: 0012-7094
DOI: 10.1215/s0012-7094-03-11826-8