Geodesic
Square-free Values of Decomposable Forms
Canadian journal of mathematics, Tome 70 (2018) no. 6, pp. 1390-1415

Voir la notice de l'article provenant de la source Cambridge University Press

In this paper we prove that decomposable forms, or homogeneous polynomials $F\left( {{x}_{1}}\,,\,.\,.\,.\,,\,{{x}_{n}} \right)$ with integer coefficients that split completely into linear factors over $\mathbb{C}$ , take on infinitely many square-free values subject to simple necessary conditions, and they have $\text{deg}\,f\,\le \,2n\,+\mid 2$ for all irreducible factors $f$ of $F$ . This work generalizes a theorem of Greaves.
DOI : 10.4153/CJM-2017-060-4
Mots-clés : 11B05, square-free value, decomposable form, Selberg sieve 
Xiao, Stanley Yao. Square-free Values of Decomposable Forms. Canadian journal of mathematics, Tome 70 (2018) no. 6, pp. 1390-1415. doi: 10.4153/CJM-2017-060-4
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[1] [1] Bhargava, M. and Shankar, A., Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves. Ann. of Math. 181 (2015), 191–242. http://dx.doi.org/10.4007/annals.2015.181.13 Google Scholar

[2] [2] Bhargava, M., Shankar, A., and Wang, X., Squarefree values of polynomial discriminants I. arxiv:1 611.09806 Google Scholar

[3] [3] Browning, T. D., Quantitative arithmetic on protective varieties. Progress in Mathematics, 277, Birkhâuser Verlag, Basel, 2009. Google Scholar

[4] [4] Ekedahl, T., An infinite version of the Chinese remainder theorem. Comment. Math. Univ. St. Paul. 40 (1991), 53–59. Google Scholar

[5] [5] Erdôs, P., Arithmetical properties of polynomials. J. London Math. Soc. 28 (1953), 416–425. http://dx.doi.Org/10.1112/jlms/s1-28.4.416 Google Scholar

[6] [6] Estermann, T., Einige Satze uber quadratfeie Zahlen. Math. Ann. 105 (1931), 653–662. http://dx.doi.Org/10.1007/BF01455836 Google Scholar

[7] [7] Granville, A., ABC allows us to count squarefrees. Internat. Math. Res. Notices 9 (1998), 991–1009. Google Scholar

[8] [8] Greaves, G., Power-free values of binary forms. Quart. J. Math Oxford Ser. (2) 43 (1992), 45–65. http://dx.doi.Org/10.1093/qmath/43.1.45 Google Scholar

[9] [9] Heath-Brown, D. R., Diophantine approximation with squarefree numbers. Math. Z. 187 (1984), 335–344. http://dx.doi.org/10.1007/BF01161951 Google Scholar

[10] [10] Helfgott, H. A., Power-free values, large deviations, and integer points on irrational curves. J. Théor. Nombres Bordeaux 19 (2007), 433–472. http://dx.doi.Org/10.58O2/jtnb.596 Google Scholar

[11] [11] Hooley, C., On the power free values of polynomials. Mathematika 14 (1967), 21–26. http://dx.doi.Org/10.1112/S002557930000797X Google Scholar

[12] [12] Hooley, C., On the power-free values of polynomials in two variables. In: Analytic number theory, Cambridge University Press, Cambridge, 2009, pp. 235–266. Google Scholar

[13] [13] Hooley, C., On the power-free values of polynomials in two variables. II. J. Number Theory 129 (2009), 1443–1455. http://dx.doi.Org/10.1 01 6/j.jnt.2 008.1 2.006 Google Scholar

[14] [14] Huxley, M. N. and Nair, M., Power free values of polynomials. III. Proc. London Math. Soc. 41 (1980), 66–82. http://dx.doi.Org/10.1112/plms/s3-41.1.66 Google Scholar

[15] [15] Lang, S. and Weil, A., Number of points of varieties over finite fields. Amer. J. Math. 76 (1954), 819–827. http://dx.doi.org/! 0.2307/2372655 Google Scholar

[16] [16] Maynard, J., Primes represented by incomplete norm forms. arxiv:1507.05080 Google Scholar

[17] [17] Poonen, B., Squarefree values of multivariate polynomials. Duke Math. J. 118 (2003), 353–373. http://dx.doi.org/10.1215/S0012-7094-03-11826-8 Google Scholar

[18] [18] Reuss, T., Power-free values of polynomials. Bull. London Math. Soc. 47 (2015), no. 2, 270–284. http://dx.doi.Org/10.1112/blms/bdu11 6 Google Scholar

[19] [19] Ricci, G., Riecenche aritmetiche sui polynomials. Rend. Circ. Mat. Palermo 57 (1933), 433–475. Google Scholar

[20] [20] Rosser, J. B. and Schoenfeld, L., Approximate formulas for some functions of prime numbers. Illinois J. Math. 6 (1962), 64–94. Google Scholar

[21] [21] Schmidt, W. M., The number of solutions of norm form equations. Trans. Amer. Math. Soc. 317 (1990), 197–227. http://dx.doi.org/10.1090/S0002-9947-1990-0961596-2 Google Scholar

[22] [22] Vaaler, J. D., Some extremal functions in Fourier analysis. Bull. Amer. Math. Soc. 12 (1985), 183–216. http://dx.doi.org/10.1090/S0273-0979-1985-15349-2 Google Scholar

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