Problem 1048

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claims/: The 2 claim pages of Problem 1048, one per claimant's result; the problem's standing derives from them.

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Statement. If $f\in \mathbb{C}[x]$ is a monic polynomial with all roots satisfying $\lvert z\rvert \leq r$ for some $r<2$, then must

$$\{ z: \lvert f(z)\rvert <1\}$$

have a connected component with diameter $>2-r$?

Status. The site's label is DISPROVED (LEAN). Pommerenke's 1961 example $z^n-r^n$, whose sublevel set for $1<r<2$ has $n$ components of diameter tending to $0$, is the accepted disproof, refuting the question for every $1<r<2$ (Pommerenke's claim page). The same paper's Theorem 3 gives the affirmative answer for $0<r\le1$ for the closed set $\{|f|\le1\}$ of Pommerenke's restatement; for $0<r\le1/2$ the bound carries over to the problem's open set, while for $1/2<r\le1$ the paper bounds the closed set only (both observations are recorded on the claim page). The degenerate case $r=0$ fails the strict inequality: $z^n$ has the open unit disc, of diameter exactly $2$, as its sublevel set. The label's qualifier (LEAN) refers to Alexeev's formalization of Pommerenke's example, which its author reports as verified by Lean, written by Aristotle and linked from that page. Aristotle's separate disproof with $z^{10}-2$, published in Lean by Alexeev, is a second accepted disproof: this corpus built and audited a later revision of its development (Alexeev's claim page).

Source. erdosproblems.com/1048, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1048, https://www.erdosproblems.com/1048.

References.

• [EHP58] Erdős, P. and Herzog, F. and Piranian, G., Metric properties of polynomials. J. Analyse Math. (1958), 125-148.
• [Po61] Pommerenke, Ch., On metric properties of complex polynomials. Michigan Math. J. 8 (1961), no. 2, 97--115, doi:10.1307/mmj/1028998561; the $z^n-r^n$ example for $1<r<2$, printed p. 98; Theorem 3 with its Remark 3 for $0<r\le1$, p. 99. Library home: pommerenke_1961_metric_properties_complex_polynomials (the publisher's open-access scan; result pages example_p98, theorem_3).

Formalization. Statement in formal-conjectures: at the linked commit the theorem erdos_1048 is left at sorry and its formal_proof attribute names the v4.29.1 file of Alexeev's formalization of Pommerenke's example, whose earlier revision is linked from Pommerenke's claim page. Aristotle's $z^{10}-2$ development Erdos1048b in the same repository, in a later revision that this corpus built and audited, is linked from Alexeev's claim page.

Progress

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Known Results

Not yet compiled.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• pommerenke_1961_metric_properties_complex_polynomials
• pommerenke_1961_metric_properties_complex_polynomials / example_p98
• pommerenke_1961_metric_properties_complex_polynomials / theorem_3
• erdos_1958_metric_properties_polynomials
• erdos_1958_metric_properties_polynomials / problem_7