Problem 1043

Let $f\in \mathbb{C}[x]$ be a monic non-constant polynomial. Must there exist a straight line $\ell$ such that the projection of

\{ z: \lvert f(z)\rvert\leq 1\}

onto $\ell$ has measure at most $2$ ?

Source: erdosproblems.com/1043

Analysis

Wiki page

Status

SolvedDisproved

An accepted solution exists. The statement is false.

DISPROVED (LEAN) on the site. Pommerenke's 1961 example, a monic polynomial whose set $\{|f|\le1\}$ projects onto every line to measure above $2.386$ , is the accepted disproof (Pommerenke's claim page (1961)); the Lean qualifier of the site's label refers to Alexeev's Lean disproof by the different polynomial $z^{16}-1$ , found by Aristotle, a later revision of which this corpus built and audited, so it is a second accepted disproof (Alexeev's claim page (2025)). Pommerenke also proved that some line always receives a projection of measure below $3.30$ .

EHP58: Erdős, P. and Herzog, F. and Piranian, G., Metric properties of polynomials. J. Analyse Math. (1958), 125-148.
Po59: Pommerenke, Ch., On some problems by Erdős, Herzog and Piranian. Michigan Math. J. 6 (1959), no. 3, 221--225, DOI 10.1307/mmj/1028998227. Theorem 1, p. 221, answers the second part of Problem 10 of [EHP58] in the negative: a lemniscate $|f(z)|=1$ with a point of support $z_0$ such that $|z-z_0|<2$ on the whole lemniscate. The first part of that problem, this page's projection question, is not treated in the paper; the site's commentary attributes the negative answer to [Po61] using Pommerenke's previous work under the key [Po59], which its reference record resolves to this note; the thread post of 12 October 2025 that the commentary answers names that work as the Math. Ann. 139 (1959) paper on the capacity of plane continua. The 1961 paper cites this note for Problem 10b (Theorem 2, p. 98) and for the width and diameter bounds of p. 109, as Pommerenke's claim page records. Library home: Pommerenke 1959 (filed 2026-09-22) and its Pommerenke 1959, Theorem 1 page.
Po61: Pommerenke, Ch., On metric properties of complex polynomials. Michigan Math. J. 8 (1961), no. 2, 97--115, doi:10.1307/mmj/1028998561; the lemniscate set with every projection of measure above $2.386$ , printed p. 103, with Theorem 7 (p. 103) and Theorem 6 (p. 102). Library home: Pommerenke 1961 (filed 2026-09-22; result page Pommerenke 1961, Unnumbered example, p. 103).

Problem 1043

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