Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-23T02:30:05.602Z Has data issue: false hasContentIssue false
  • English
  • Français

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 [email protected]
Stanley Yao Xiao
Affiliation:
Department of Mathematics, University of Toronto, Bahen Centre, 40 St. George St., Room 6290, Toronto, Ontario, CanadaM5S 2E4 email [email protected]
Get access

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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