On Schur-szegö Composition of Polynomials

Schur-Szegö composition of two polynomials of degree less or equal than a given positive integer n introduces an interesting semigroup structure on polynomial spaces and is one of the basic tools in the analytic theory of polynomials, see [4]. In the present paper we add several (apparently) new aspects to the previously known properties of this operation. Namely, we show how it interacts with the stratification of polynomials according to the multiplicities of their zeros and present the induced semigroup structure on the set of all ordered partitions of n. The Schur-Szegö composition of two polynomials P (x) = ∑n i=0 C i naix i and Q(x) = ∑n i=0 C i nbix i is given by P ∗ Q(x) = ∑n i=0 C i naibix , see e.g. [5]. Let Poln denote the linear space of all polynomials in x of degree at most n. In what follows we always use its standard monomial basis B := (x, x, . . . , 1). To any polynomial P ∈ Poln one can associate the operator TP which acts diagonally in B and is uniquely determined by the condition: TP (1 + x) n = P (x). Obviously, for P (x) = C na0 + C 1 na1x + · · ·+ C n nanx n one has TP (x ) = ai, i = 0, 1, . . . , n. Given P as above we refer to the sequence {ai} as to the diagonal sequence of P . Any two such operators TP and TQ commute and their product TPTQ corresponds in the above sense exactly to the Schur-Szegö composition P ∗Q. The famous composition theorem of Schur and Szegö (see original [5] and e.g. §3.4 of [4] or §2 of [1]) reads: Theorem 1. Given any linear-fractional image K of the unit disk containing all the roots of P one has that any root of P ∗Q is the product of some root of Q by −γ where γ ∈ K. Geometric consequences of Theorem 1, in particular, Proposition 2, can be found in § 5.5 of [4]. A polynomial P ∈ Poln is called hyperbolic if all its roots are real. Denote byHypn ⊂ Poln the set of all hyperbolic polynomials and byHyp + n ⊂ Hypn (resp. Hyp−n ⊂ Hypn) the set of all hyperbolic polynomials with all positive (resp. all negative) roots. Denote by Hu,v,w ⊂ Hypn (where u, v, w ∈ N∪0, u+v+w = n) the set of all hyperbolic polynomials with u negative and w positive roots and a v-fold zero root. Proposition 2 (Theorem 5.5.5 and Corollary 5.5.10 of [4]). If P,Q ∈ Hypn and if Q ∈ Hypn or Q ∈ Hyp − n , then P ∗ Q ∈ Hypn. Moreover, all roots of P ∗ Q lie in [−M,−m] where M is the maximal and m is the minimal pairwise product of roots of P and Q. A diagonal sequence, (or an operator T : Poln → Poln acting diagonally in B) is called a finite multiplier sequence (FMS), see [3], if it sends Hypn into Hypn. The set Mn of all FMS is a semigroup. For the following characterization of FMS see [2], Theorem 3.7 or [1], Theorem 3.1. Theorem 3. For T ∈End(Pol k ) the following two conditions are equivalent: Date: February 2, 2008. 1991 Mathematics Subject Classification. 12D10.