Erdos 1973 remark polynomials transfinite diameter

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lemma_p23: For a bounded, closed and connected set D of transfinite diameter 1 - c with 0 < c < 1 there is a monic polynomial whose degree depends only on c and whose modulus is below one half on D.

theorem_p23: For a bounded, closed and connected set D of transfinite diameter 1 - c with 0 < c < 1, every monic polynomial with all zeros in D has modulus below one on some disc of a radius depending only on c, not on the degree.

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P. Erdős, E. Netanyahu: A remark on polynomials and the transfinite diameter, Israel J. Math. 14 (1973), 23--25 (MR 47 #7006; Zentralblatt 259.30004).

For f(z) = prod_{v=1}^n (z-z_v) let E(f) be the set where |f(z)| < 1. The single theorem (p. 23) states that if D is a bounded, closed, connected set of transfinite diameter d(D) = 1-c with 0 < c < 1, and all zeros z_v lie in D, then E(f) contains a disc of radius ρ = ρ(c) depending only on c and not on n. The proof rests on a lemma (pp. 23--24), proved by contradiction using Fekete's mapping theorem and a normal-family argument on the exterior conformal maps of a hypothetical sequence of sets D_n, which produces a polynomial P(z) = z^m + ... of degree m = m(c) depending only on c with |P(z)| < 1/2 on D; perturbing the zeros t_i of P within discs H_i of radius ρ and comparing the product of the values |f(s_i)| then shows that |f| < 1 throughout one of the discs H_i. The authors note that a weaker result is Theorem 6 of Erdős, Herzog and Piranian, that their argument is only an existence proof, and that a numerical estimate for ρ and for the degree m would be of interest. They remark (p. 25) that the theorem fails for disconnected D, that it implies a lower bound depending only on c for the area of the closed sublevel set |f(z)| <= 1, and that for D of transfinite diameter 1 this area can perhaps be made smaller than any ε once n is large, which Erdős, Herzog and Piranian prove for the unit circle and the interval (-2, 2), the general case being open. A last remark (p. 25) raises the maximum number of components of the closed sublevel set |f(z)| <= 1, recalls the maximum n-1 for the unit circle from Erdős, Herzog and Piranian, and says without proof that for the interval (-2, 2) the set E(f) can have n components.

Source: https://users.renyi.hu/~p_erdos/1973-01.pdf. No notice is printed on any of the three pages (23--25), and the publisher's page was not consulted; the Crossref record for DOI 10.1007/bf02761531, read 2026-10-02, names only the publisher's own terms, Springer's text-and-data-mining license (springer.com/tdm), and no open license, every other right reserved.

Results. Theorem, p. 23 (a disc of radius $\rho(c)$ inside every sublevel set $|f|<1$, with the p. 25 remarks); Lemma, pp. 23--24 (a monic polynomial of degree $m(c)$ below $\tfrac12$ on $D$).

Bears on.

• E1040: the problem page cites the paper as [ErNe73]. For bounded, closed, connected sets of transfinite diameter $1-c$ with $0<c<1$ the theorem puts a disc of radius $\rho(c)$, depending only on $c$, inside every sublevel set $\{z:|f(z)|<1\}$, and the paper notes the resulting lower bound, depending only on $c$, for the area of the closed set $\{z:|f(z)|\le1\}$ (p. 25). For transfinite diameter $1$ it only suggests that this area can be made arbitrarily small for large $n$, records that Erdős, Herzog and Piranian proved this for the unit circle and the interval $(-2,+2)$, and calls the general case open.
• E1042: the p. 25 remark on the number of components of the closed set $\{z:|f(z)|\le1\}$ records the unit-circle maximum $n-1$ from Erdős, Herzog and Piranian and states without proof that for zeros in the interval $(-2,+2)$ the set $E(f)$ can have $n$ components; it proves nothing on the problem.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.