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DENSITY OF POWER-FREE VALUES OF POLYNOMIALS

Part of: Multiplicative number theory Diophantine equations

Published online by Cambridge University Press:  14 August 2019

Kostadinka Lapkova  and
Stanley Yao Xiao
Kostadinka Lapkova
Affiliation:
Institute of Analysis and Number Theory, Graz University of Technology, Kopernikusgasse 24/II, 8010 Graz, Austria email lapkova@math.tugraz.at
Stanley Yao Xiao
Affiliation:
Department of Mathematics, University of Toronto, Bahen Centre, 40 St. George St., Room 6290, Toronto, Ontario, CanadaM5S 2E4 email stanley.xiao@math.toronto.edu
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Abstract

We establish asymptotic formulae for the number of $k$-free values of square-free polynomials $F(x_{1},\ldots ,x_{n})\in \mathbb{Z}[x_{1},\ldots ,x_{n}]$ of degree $d\geqslant 2$ for any $n\geqslant 1$, including when the variables are prime, as long as $k\geqslant (3d+1)/4$. This generalizes a work of Browning.

MSC classification

Primary: 11N32: Primes represented by polynomials; other multiplicative structure of polynomial values
Secondary: 11D45: Counting solutions of Diophantine equations
Type
Research Article
Information
Mathematika , Volume 65 , Issue 4 , 2019 , pp. 1038 - 1050
Copyright
Copyright © University College London 2019 

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References

Bhargava, M., The geometric sieve and the density of squarefree values of invariant polynomials. Preprint, 2014, arXiv:1402.0031 [math.NT].Google Scholar
Bhargava, M. and Shankar, A., Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves. Ann. of Math. (2) 181 2015, 191–242.10.4007/annals.2015.181.1.3Google Scholar
Bhargava, M., Shankar, A. and Wang, X., Squarefree values of polynomial discriminants I. Preprint, 2016, arXiv:1611.09806 [math.NT].Google Scholar
Browning, T. D., Power-free values of polynomials. Arch. Math. (2) 96 2011, 139–150.Google Scholar
Erdős, P., Arithmetical properties of polynomials. J. Lond. Math. Soc. 28 1953, 416–425.10.1112/jlms/s1-28.4.416Google Scholar
Filaseta, M., Powerfree values of binary forms. J. Number Theory 49 1994, 250–268.10.1006/jnth.1994.1092Google Scholar
Greaves, G., Power-free values of binary forms. Q. J. Math. 43 1992, 45–65.Google Scholar
Heath-Brown, D. R., The density of rational points on curves and surfaces. Ann. of Math. (2) 155 2002, 553–598.10.2307/3062125Google Scholar
Heath-Brown, D. R., Counting rational points on algebraic varieties. In Analytic Number Theory (Lecture Notes in Mathematics 1891 ), Springer (Berlin, 2006), 51–95.10.1007/978-3-540-36364-4_2Google Scholar
Hooley, C., On the power free values of polynomials. Mathematika 14 1967, 21–26.10.1112/S002557930000797XGoogle Scholar
Hooley, C., On the power-free values of polynomials in two variables. In Analytic Number Theory, Cambridge University Press (2009), 235–266.Google Scholar
Hooley, C., On the power-free values of polynomials in two variables: II. J. Number Theory 129 2009, 1443–1455.10.1016/j.jnt.2008.12.006Google Scholar
Lapkova, K., On the k-free values of the polynomial xy k + C . Acta Math. Hungar. 149(1) 2016, 190–207.10.1007/s10474-016-0594-1Google Scholar
Le Boudec, P., Power-free values of the polynomial t 1… t r - 1. Bull. Aust. Math. Soc. 85 2012, 154–163.Google Scholar
Poonen, B., Squarefree values of multivariable polynomials. Duke Math. J. 118 2003, 353–373.10.1215/S0012-7094-03-11826-8Google Scholar
Salberger, P., Counting rational points on projective varieties. Preprint, 2009.Google Scholar
Xiao, S. Y., Power-free values of binary forms and the global determinant method. Int. Math. Res. Not. IMRN 2017(16) 2017, 5078–5135.Google Scholar
Xiao, S. Y., Square-free values of decomposable forms. Canad. J. Math. 70(6) 2018, 1390–1415.10.4153/CJM-2017-060-4Google Scholar