A Number Theoretic Problem on the Distribution of Polynomials with Bounded Roots
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چکیده
Let E d denote the set of coe cient vectors (a1, . . . , ad) 2 Rd of contractive polynomials xd + a1x 1 + · · ·+ ad 2 R[x] that have exactly s pairs of complex conjugate roots and let v d = d(E (s) d ) be its (d-dimensional) Lebesgue measure. We settle the instance s = 1 of a conjecture by Akiyama and Pethő, stating that the ratio v d /v (0) d is an integer for all d 2s. Moreover we establish the surprisingly simple formula v d /v (0) d = (Pd(3) 2d 1)/4, where Pd(x) are the Legendre polynomials. Dedicated to Prof. Dominique Foata on the occasion of his 80 th birthday.
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تاریخ انتشار 2015