Polynomials with zeros and small norm on curves∗

This note considers the problem how zeros lying on the boundary of a domain influence the norm of polynomials (under the normalization that their value is fixed at a point). It is shown that k zeros raise the norm by a factor (1 + ck/n) (where n is the degree of the polynomial), while k excessive zeros on an arc compared to n times the equilibrium measure raise the norm by a factor exp(ck/n). These bounds are sharp, and they generalize earlier results for the unit circle which are connected to some constructions in number theory. Some related theorems of Andrievskii and Blatt will also be strengthened. 1 Results Let C1 = {z |z| = 1} be the unit circle. The paper [12] discussed monic polynomials with prescribed zeros on C1 having as small norm as possible. The problem goes back to Turán’s power sum method in number theory, in connection with which G. Halász [6] showed that there is a polynomial Qn(z) = z + · · · with a zero at 1 and of norm ∥Qn∥C1 ≤ exp(2/n), where ∥ · ∥K denotes supremum norm on the compact setK. See [7] for the smallest possible norm for such a polynomial. Halász’ result implies that if Z1, Z2, . . . , Zkn are arbitrary kn < n/2 points on the unit circle, then there is a Pn = z n + · · · which has a zero at each Zj and has norm ∥Pn∥C1 ≤ exp(4k n/n) (1) It was shown in [12, Theorem 1] that, in general, one cannot have smaller norm, namely there is a constant c > 0 with the following property: for any monic polynomials Pn(z) = z n + · · · (i) if Pn has k zeros (counting multiplicity) on C1, then ∥Pn∥C1 ≥ 1 + c(k/n), (ii) if Pn has n|J |/2π + k zeros (counting multiplicity) on a subarc J = Jn of the unit circle, then ∥Pn∥C1 ≥ exp(ck/n). ∗