HILBERT SERIES OF QUADRATIC ALGEBRAS ASSOCIATED WITH PSEUDO-ROOTS OF NONCOMMUTATIVE POLYNOMIALS Israel Gelfand, Sergei Gelfand,Vladimir Retakh,

The quadratic algebras Qn are associated with pseudo-roots of noncommutative polynomials. We compute the Hilbert series of the algebras Qn and of the dual algebras Q ! n. Introduction Let P (x) = x−a1x n−1 + · · ·+(−1)an be a polynomial over a ring R. Two classical problems concern the polynomial P (x): nvestigation of the solutions of the equation P (x) = 0 and the decomposition of P (x) into a product of irreducible polynomials. In the commutative case relations between these two problems are well known: when R is a commutative division algebra, x is a central variable, and the equation P (x) = 0 has roots x1, . . . , xn, then (0.1) P (x) = (x− xn) . . . (x− x2)(x− x1). In noncommutative case relations between the two problems are highly non-trivial. They were investigated by Ore [O] and others. ([L] is a good source for references, see also the book [GLR] where matrix polynomials are considered.) More recently, some of the present authors have obtained results [GR3, GR4, W] which are important for the present work. For a division algebra R, I. Gelfand and V. Retakh [GR3, GR4] studied connections between the coefficients of P (x) and a generic set of solutions x1, . . . , xn of the equation P (x) = 0. They showed that for Typeset by AMS-TEX 1 any ordering I = (i1, . . . , in) of (1, . . . , n) one can construct elements yk , k = 1, . . . , n, depending on xi1 , . . . , xik such that a1 = y1 + y2 + · · ·+ yn,