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  • Kostadinka Lapkova (a1) and Stanley Yao Xiao (a2)


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.



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1. Bhargava, M., The geometric sieve and the density of squarefree values of invariant polynomials. Preprint, 2014, arXiv:1402.0031 [math.NT].
2. 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, 191242.10.4007/annals.2015.181.1.3
3. Bhargava, M., Shankar, A. and Wang, X., Squarefree values of polynomial discriminants I. Preprint, 2016, arXiv:1611.09806 [math.NT].
4. Browning, T. D., Power-free values of polynomials. Arch. Math. (2) 96 2011, 139150.
5. Erdős, P., Arithmetical properties of polynomials. J. Lond. Math. Soc. 28 1953, 416425.10.1112/jlms/s1-28.4.416
6. Filaseta, M., Powerfree values of binary forms. J. Number Theory 49 1994, 250268.10.1006/jnth.1994.1092
7. Greaves, G., Power-free values of binary forms. Q. J. Math. 43 1992, 4565.
8. Heath-Brown, D. R., The density of rational points on curves and surfaces. Ann. of Math. (2) 155 2002, 553598.10.2307/3062125
9. Heath-Brown, D. R., Counting rational points on algebraic varieties. In Analytic Number Theory (Lecture Notes in Mathematics 1891 ), Springer (Berlin, 2006), 5195.10.1007/978-3-540-36364-4_2
10. Hooley, C., On the power free values of polynomials. Mathematika 14 1967, 2126.10.1112/S002557930000797X
11. Hooley, C., On the power-free values of polynomials in two variables. In Analytic Number Theory, Cambridge University Press (2009), 235266.
12. Hooley, C., On the power-free values of polynomials in two variables: II. J. Number Theory 129 2009, 14431455.10.1016/j.jnt.2008.12.006
13. Lapkova, K., On the k-free values of the polynomial xy k + C . Acta Math. Hungar. 149(1) 2016, 190207.10.1007/s10474-016-0594-1
14. Le Boudec, P., Power-free values of the polynomial t 1… t r - 1. Bull. Aust. Math. Soc. 85 2012, 154163.
15. Poonen, B., Squarefree values of multivariable polynomials. Duke Math. J. 118 2003, 353373.10.1215/S0012-7094-03-11826-8
16. Salberger, P., Counting rational points on projective varieties. Preprint, 2009.
17. Xiao, S. Y., Power-free values of binary forms and the global determinant method. Int. Math. Res. Not. IMRN 2017(16) 2017, 50785135.
18. Xiao, S. Y., Square-free values of decomposable forms. Canad. J. Math. 70(6) 2018, 13901415.10.4153/CJM-2017-060-4
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  • Kostadinka Lapkova (a1) and Stanley Yao Xiao (a2)


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