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Squarefree values of multivariable polynomials. (English) Zbl 1047.11021

Let \(A\) be a given ring and \(f\in A[x_1,\dots ,x_n]\). The author seeks to confirm the heuristic that the density of squarefree values taken by the polynomial \(f\) (that is \(f(x)\) for \(x\in A^n\) in some appropriate expanding domain) is equal to the product of the density of squarefree values taken by the polynomial mod \(p\), where the product runs over the (finite) primes \(p\) in the ring.
This is proved in the ring \(A=\mathbb{F}_q[t]\), generalizing Ramsay’s result for \(n=1\) [K. Ramsay, Int. Math. Res. Not. 1992, No. 4, 97–102 (1992; Zbl 0760.11037)], as well as getting similar results in certain other rings of functions. The key ingredient is the fact that \(\mathbb{F}_q[t]\) has an \(\mathbb{F}_q[t^p]\)-linear derivation.
The author also proves a weak version of this in the ring \(A=\mathbb{Z}\) assuming the \(abc\)-conjecture, generalizing the reviewer’s result for \(n=1\) [Int. Math. Res. Not. 1998, No. 19, 991–1009 (1998; Zbl 0924.11018)] by fibering. However since the proof in the \(n=1\) case gives estimates that are not too uniform (due to the use of Belyi’s theorem), when the author fibers he must essentially fix \(n-1\) variables and allow the last to run to \(\infty\), so that the “expanding domain” for which the result is thus proved is not the most natural. Thus proving the result when the variables lie in an expanding \(n\)-dimensional cube, even under the assumption of the \(abc\)-conjecture, remains the central open problem.
It is also proved that the density of values at which two polynomials \(f\) and \(g\) in \(n\) variables are coprime is equal to the product of the density of values at which the two polynomials are not both divisible by \(p\), asymptotically, for \(A=\mathbb{F}_q[t]\) or \(\mathbb{Z}\). Finally the author answers a question of the reviewer, showing that the number of distinct values of \(f(m)\) mod \(Q/Q^2\) with \(m\leq M\) is \(\sim M\), assuming the \(abc\)-conjecture, if \(f(x)\) has no repeated factors and has degree \(\geq 2\) (a result that was simultaneously proved, independently, by P. Cutter, T. J. Tucker and the reviewer in [Can. Math. Bull. 46, 71–79 (2003; Zbl 1044.11087)]). Finally the author asks for the analagous result when \(n\geq 2\).
In summary, the author proves some important results on the surprisingly difficult subject of counting squarefree values of polynomials, and helps delineate what are the key challenges in the area.

MSC:

11C08 Polynomials in number theory
11D75 Diophantine inequalities
11N32 Primes represented by polynomials; other multiplicative structures of polynomial values
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References:

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