Let E(s) d denote the set of coecient vectors (a1, . . . , ad) 2 Rd of contractive polynomials xd + a1xd1 + · · · + ad 2 R[x] that have exactly s pairs of complex conjugate roots and let v(s) d = d(E(s) d ) be its (d-dimensional) Lebesgue measure. We settle the instance s = 1 of a conjecture by Akiyama and Peth?o, stating that the ratio v(s) d /v(0) d is an integer for all d 2s. Moreover we establish the surprisingly simple formula v(1) 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.
展开▼
