On the number of reducible polynomials of bounded naive height |
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Authors: | Artūras Dubickas |
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Institution: | 1. Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, 03225, Vilnius, Lithuania 2. Institute of Mathematics and Informatics, Vilnius University, Akademijos 4, 08663, Vilnius, Lithuania
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Abstract: | We prove an asymptotical formula for the number of reducible integer polynomials of degree d and of naive height at most T when \({T \to \infty}\) . The main term turns out to be of the form \({\kappa_d T^d}\) for each \({d \geq 3}\) , where the constant \({\kappa_d}\) is given in terms of some infinite Dirichlet series involving the volumes of symmetric convex bodies in \({\mathbb{R}^d}\) . For d = 2, we prove that there are asymptotically \({\kappa_2 T^2 \,\text{log} T}\) of such polynomials, where \({\kappa_2:=6(3\sqrt{5}+2\,\text{log} (1+\sqrt{5}) -2 \,\text{log}\, 2)/\pi^2}\) . Earlier results in this direction were given by van der Waerden, Pólya and Szegö, Dörge, Chela, and Kuba. |
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