2022-01-27T17:53:47Zhttps://tsukuba.repo.nii.ac.jp/oaioai:tsukuba.repo.nii.ac.jp:000346832021-03-01T21:19:48ZOn the distribution of polynomials with bounded roots, I. Polynomials with real coefficients秋山, 茂樹AKIYAMA, ShigekiPETHŐ, AttilaLet v(s)d denote the set of coefficient vectors of contractive polynomials of degree d with 2s non-real zeros.We prove that v(s)d can be computed by a multiple integral, which is related to the Selberg integral and its generalizations. We show that the boundary of the above set is the union of finitely many algebraic surfaces. We investigate arithmetical properties of v(s)d and prove among others that they are rational numbers. We will show that within contractive polynomials, the ‘probability’ of picking a totally real polynomial decreases rapidly when its degree becomes large.journal articleThe Mathematical Society of Japan2014-07application/pdfJournal of the Mathematical Society of Japan3669279490025-5645AA0070177Xhttps://tsukuba.repo.nii.ac.jp/record/34683/files/JMSJ_66-3.pdfeng10.2969/jmsj/06630927© 2014 The Mathematical Society of Japan