Problem 1215

../

claims/: The 1 claim page of Problem 1215, one per claimant's result; the problem's standing derives from them.

---

Statement. Does there exist a constant $C$ such that for every polynomial $P$ with $P(0)=1$, all of whose roots are on the unit circle, there exists a path in

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

which connects $0$ to the unit circle of length at most $C$?

Statement (corrected). Does there exist a constant $C$ such that for every nonconstant polynomial $P$ with $P(0)=1$, all of whose roots are on the unit circle, there exists a path in

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

everywhere except at $z=0$, which connects $0$ to the unit circle of length at most $C$?

Notes. The site's wording fails for every $P$. Since $P(0)=1$, the point $0$ is not in $\{z:\lvert P(z)\rvert<1\}$, so no path inside that set starts at $0$, and the answer is no for every $C$, trivially; the smallest instance is $P(z)=1-z$. It fails a second way at the constant polynomial $P=1$, which has $P(0)=1$ and no roots: its set is empty, so no path exists even with the starting point exempt. The change inserts "everywhere except at $z=0$" after the set and "nonconstant" before "polynomial"; nothing else changes. The first insertion is in the posers' own words. Erdős, Herzog and Piranian [EHP55, §1, p. 347] (library card: erdos_1955_polynomials_whose_zeros_lie_unit_circle) state Cohen's theorem as giving a path from the origin to the unit circle on which "the inequality $|P| < 1$ holds everywhere except at $z = 0$", and two paragraphs later ask, in connection with their Theorem 1, whether a universal constant $L$ exists such that for every polynomial (1) "the inequality $|P| < 1$ holds on a path which connects the origin to $C$ and has length at most $L$", their $C$ being the unit circle; the question repeats the phrase of Cohen's theorem, and the posers report that Mac Lane answered it in the negative. The site's commentary states Cohen's theorem in the words of the site's question, a path in the set that connects $0$ to the unit circle, which is true only with the starting point exempt. The second insertion is the corpus's own correction. It excludes exactly the polynomials of degree $0$, at which no path can meet the conclusion. At degree $1$, $P(z)=1-z/\omega$ with $\lvert\omega\rvert=1$, the radius from $0$ to $\omega$ has $\lvert P\rvert<1$ except at $0$ and length $1$, and at every degree $n\ge1$ Cohen's theorem gives a path, so no other degree fails this way. The defect is already in the posers' question, which states the exemption for Cohen's path but not again in the question, and says nothing of degree $0$. The one result about the site's wording is the trivial answer above, the corpus's own observation, published nowhere else; it is credited here and counts for nothing. Mac Lane's theorem answers the corrected Statement no, and the problem's label and standing judge the corrected Statement.

Status. Disproved, the site's label, which describes the corrected Statement. Mac Lane proved that for every compact set inside a simply connected subdomain of the open disc avoiding $0$, all large degrees admit such a polynomial with modulus above $2$ on the set, and a spiral forces arbitrarily long paths; the accepted claim is Mac Lane's unbounded path length, refereed and credited by the site's curator.

Source. erdosproblems.com/1215, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1215, https://www.erdosproblems.com/1215.

References.

• [Co52] P. Cohen, Modulus of an analytic function. American Mathematical Monthly (1952), 704-705.
• [EHP55] Erdős, P. and Herzog, F. and Piranian, G., Polynomials whose zeros lie on the unit circle. Duke Math. J. (1955), 347-351.
• [Ma53] Mac Lane, Gerald R., On a conjecture of Erdös, Herzog, and Piranian. Michigan Math. J. (1953/54), 147-148.

Formalization. No statement file in formal-conjectures. An external Lean file that declares itself a formalization of Mac Lane's solution is a formalization link on Mac Lane's claim page; this corpus has not built it.

Progress

Not yet compiled.

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.

• erdos_1955_polynomials_whose_zeros_lie_unit_circle
• erdos_1955_polynomials_whose_zeros_lie_unit_circle / question_p347
• erdos_1955_polynomials_whose_zeros_lie_unit_circle / theorem_1