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Abstract: Given f Z[x 1 , . . . , x n ], we compute the density of x such that f(x) is squarefree, assuming the abc conjecture. Given f, g Z[x 1 , . . . , x n ], we compute unconditionally the density of x 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, 2, . . . , n}) in Q # /Q #2 # {0}. 1. (Update)
Context of citations to this paper: More
...main theorems for regular quasiprojective schemes over Spec Z. The abc conjecture is needed to apply a multivariable generalization [Poo01] of A. Granville s result [Gra98] about squarefree values of polynomials. For some open questions, see Sections 4 and 5.7, and also...
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Bertini Theorems over Finite Fields - Poonen (2003) (Correct)
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3.4: Squarefree Values of Multivariable Polynomials - Poonen (2003) (Correct)
0.4: Powerfree Values of Binary Forms - Filaseta (1994) (Correct)
0.3: Duality In Algebra And Topology - Dwyer Greenlees And (Correct)
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0.7: ABC Allows Us to Count Squarefrees - Granville (1998) (Correct)
0.5: The Number Of Fields Generated By The Square Root Of.. - Cutter, Granville.. (2002) (Correct)
0.5: Squarefree Values Of Polynomials And The Abc-Conjecture - Browkin, Filaseta.. (1997) (Correct)
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2: The square-free sieve and the rank of elliptic curves (context) - Gouvea, Mazur - 1991
2: Space filling curves over finite fields (context) - Katz - 1999
2: the power free values of polynomials (context) - Hooley - 1967
BibTeX entry: (Update)
Bjorn Poonen, Squarefree values of multivariable polynomials, in preparation, 2001. http://citeseer.ist.psu.edu/article/poonen01squarefree.html More
@misc{ poonen01squarefree,
author = "B. Poonen",
title = "Squarefree values of multivariable polynomials",
text = "Bjorn Poonen, Squarefree values of multivariable polynomials, in preparation,
2001.",
year = "2001",
url = "citeseer.ist.psu.edu/article/poonen01squarefree.html" }
Citations (may not include all citations):439 An introduction to the theory of numbers (context) - Hardy, Wright - 1979
103 Introduction to commutative algebra (context) - Atiyah, Macdonald - 1969
13 the power free values of polynomials (context) - Hooley - 1967
12 Applications of sieve methods to the theory of numbers (context) - Hooley - 1976
7 The square-free sieve and the rank of elliptic curves (context) - Gouvea, Mazur - 1991
6 Translated from the French and edited by Martin Brown from n.. (context) - Serre, the et al. - 1997
6 ABC allows us to count squarefrees - Granville - 1998
5 Bertini theorems over finite fields - Poonen - 2002
4 Power-free values of binary forms (context) - Greaves - 1992
3 Translated from the Japanese by M (context) - Matsumura, theory et al. - 1989
2 The number of fields generated by the square root of values .. - Cutter, Granville et al. - 2002
2 Square-free values of polynomials in one variable over funct.. (context) - Ramsay - 1992
2 An infinite version of the Chinese remainder theorem (context) - Ekedahl - 1991
2 Counting elliptic surfaces over finite fields (context) - de Jong - 2001
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