Abstract. On a compact subset of the complex plane the supremum norm of a polynomial of degree $n$ with leading coefficient $1$ must be at least the $n$-th power of the logarithmic capacity of the set. In general, nothing more can be said, but if the polynomial also has zeros on the outer boundary, then those zeros may raise the minimal norm. The paper quantifies how much zeros on the boundary raise the norm on sets bounded by finitely many smooth Jordan curves. For example, $k_n$ zeros results in a factor $(1+ck_n/n)$, while $k_n$ excessive zeros on a subarc of the boundary compared to the expected value based on the equilibrium measure introduces an exponential factor $\exp(ck_n^2/n)$. The results are sharp, and they are related to Turán's power-sum method in number theory. It is also shown by an example that the smoothness condition cannot be entirely dropped.

DOI: 10.14232/actasm-014-323-9

AMS Subject Classification (1991): 42C05, 31A15

Keyword(s): monic polynomials, minimal norm, zeros on the boundary, equilibrium measure

Received November 8, 2014. (Registered under 73/2014.)