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Source. The unnumbered Lemma, stated on pp. 23--24 and proved on p. 24, of P. Erdős and E. Netanyahu, A remark on polynomials and the transfinite diameter, Israel J. Math. 14 (1973), 23--25, DOI 10.1007/BF02761531, the edition named on the source card.

Statement

Lemma (pp. 23--24). Let $D$ be a bounded, closed and connected set whose transfinite diameter $d(D)$ equals $1-c$, where $0<c<1$ (the hypotheses of the Theorem). Then there is always a polynomial $P(z)=z^m+a_1z^{m-1}+\cdots+a_m$ whose degree $m=m(c)$ depends only on $c$ such that $|P(z)|<\tfrac12$ on $D$.

The paper adds that $\tfrac12$ may be replaced by any fixed $a$ with $0<a<1$ (p. 24). It calls its proof an existence proof and says a numerical estimate for the degree would be interesting (p. 23).

Read depth. Claims checked: the statement was read clause by clause on the page images of pp. 23--24. The proof was read in outline only and not checked step by step. Nothing here is independently reviewed.

Proof pointer

Page 24, written here in outline. The proof is by contradiction. If the lemma failed there would be bounded, closed, connected sets $D_n$, all containing $0$ and of transfinite diameter $1-c$, on which every monic polynomial bounded by $\tfrac12$ has degree at least $n$. Pass to the complement of the unbounded component of the complement of each $D_n$. By a theorem of Fekete (Math. Z. 17 (1923), 228--249) the exterior of each such set is mapped conformally onto $|\zeta|>1-c$, normalized at infinity. The inverse maps form a normal family, so a subsequence converges on $|\zeta|>1-c+\varepsilon$, with $1-c+\varepsilon<1$, to the exterior map of a set $D^*$ of transfinite diameter $1-c+\varepsilon$. The level curves of the subsequence bound domains containing the $D_{n_k}$ and converging to $D^*$, so no monic polynomial would be bounded by $\tfrac12$ on $D^*$, which contradicts Fekete's results (§§2, 3 of his paper) since $d(D^*)<1$.

Dependencies

Fekete's mapping theorem and §§2--3 of the same paper of Fekete, both cited from the paper's reference [2]; no other result of the same paper.

Bears on

• Problem 1040: the lemma is the input of the Theorem, which bounds below the radius of a disc inside every sublevel set $\{z:|f(z)|<1\}$ for zeros in such a $D$; on its own it states nothing about the area in the problem.