Pommerenke 1961 metric properties complex polynomials

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example_p103: A lemniscate set whose projection onto every line has measure greater than 2.386, obtained from the "5-Stern" through the approximation theorem, with Theorem 7's complementary bound that some projection of a capacity-1 set has measure less than 3.30; the negative answer to Problem 1043.

example_p98: For 1 < r < 2 the set where |z^n - r^n| is at most 1 has n components whose common diameter tends to 0 as n grows, so no component has diameter at least 2 - r; the negative half of the answer to Problem 1048.

theorem_1: For every l below 4 and every k there is a monic polynomial whose sublevel set has at least k components of diameter at least l; the negative answer to Problem 511 and to Problems 8 and 9 of the 1958 paper.

theorem_10: With the centroid of the zeros at 0, E is connected when the zeros lie in the disk of radius 1/sqrt 2 or in [-1, 1], and a connected E lies in the disk of radius 2 about the centroid, with the zeros inside it and sigma below sqrt 2; the connected case of Problem 509.

theorem_14: The polynomial z^p (z - a), for large p and a slightly above (1 + 1/p) p^{1/(p+1)}, has a sublevel set with two components one of which is not convex; the negative answer to Grunsky's question, Problem 1047.

theorem_16: The maximum of the ordered product of the distances among n points of diameter at most 2 is at most 2^{4(n-1)} n^n, from Theorem 15 on convex continua of capacity 1; the first general upper bound for Problem 1045, with the remark that the hull of a maximal system is nearly a disk.

theorem_3: When the zeros lie in the disk of radius r at most 1, the component of E containing 0 has diameter at least 2, more than 1/r, or more than 2 - r^2 in three ranges of r, in every case at least 2 - r; the affirmative half of the answer to Problem 1048, for the closed set.

theorem_4: When the zeros lie in the closed unit disk, E contains a disk of radius (2e)^{-1} n^{-2}, so its area is at least pi (2e)^{-2} n^{-4}; the polynomial lower bound of Problem 116 and the n^{-2} inradius bound of Problem 1039.

theorem_9: The lemniscate |f(z)| = 1 of a monic polynomial of degree n has length less than 74 n^2, the first polynomial upper bound toward the length question of Problem 114.

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Ch. Pommerenke, On metric properties of complex polynomials, Michigan Math. J. 8 (1961), no. 2, 97--115, DOI 10.1307/mmj/1028998561; received November 26, 1960 (footnote, p. 97); the author at the University of Göttingen (p. 115). Cited as [Po61] on the problem pages. The paper's [2] is Erdős, Herzog and Piranian, Metric properties of polynomials (1958), filed as erdos_1958_metric_properties_polynomials; its [9] is the author's 1959 note On some problems by Erdős, Herzog and Piranian, Michigan Math. J. 6 (1959), 221--225, cited as [Po59] on the problem pages and filed as pommerenke_1959_some_problems_erdos_herzog_piranian; its [11] is the author's On the derivative of a polynomial, Michigan Math. J. 6 (1959), 373--375; its [10] and [12] are the author's Math. Ann. papers Über die Kapazität ebener Kontinuen (139 (1959/60), 64--75) and Einige Sätze über die Kapazität ebener Mengen (141 (1960), 143--152), and its [8] is Pólya's 1928 Berlin Akademie note on the projection of a set of capacity 1. The thirteen references are printed on p. 115.

The copy read for this card is the publisher's scan of the printed article as served by Project Euclid: 20 PDF pages, printed pp. 97--115 = PDF pp. 1--19 (printed p. $n$ is PDF p. $n-96$), PDF p. 20 blank; an image-only scan with no text layer (its metadata names a 2002 conversion by c42pdf and PDFlib), so every passage below was read on the page image. Provenance: the copy was obtained free of charge on 2026-09-22 from the publisher's open-access article page on Project Euclid, from https://projecteuclid.org/journals/michigan-mathematical-journal/volume-8/issue-2/On-metric-properties-of-complex-polynomials/10.1307/mmj/1028998561.full (the article's PDF download endpoint); 1,315,023 bytes. No notice is printed on the scan's first or last page, and the publisher's article page labels the article Open Access and shows no copyright or license line (https://projecteuclid.org/journals/michigan-mathematical-journal/volume-8/issue-2/On-metric-properties-of-complex-polynomials/10.1307/mmj/1028998561.full, read 2026-10-02); the term is unstated.

Read status: claims checked for every numbered statement of the paper, Theorems 1--16 and Lemmas 1--6, and for the unnumbered passages on Problem 7 (p. 98), on the minimum projection $b$ (p. 103), on the width and diameter of a continuum (p. 109) and on Problem 13 (p. 112), each read clause by clause on the page images of PDF pp. 1--19 (printed pp. 97--115) on 2026-09-22; the footnote, the section headings and the reference list were read on the same images. The proofs of Theorem 1, of the $z^n-r^n$ example, of Lemmas 1--3, of Theorems 4 and 7, of Theorem 14 and of Theorems 15--16 (a paragraph or a page each) were read in full and their steps followed; the proofs of Theorems 3, 5, 6, 9, 10, 11, 12 and 13 and of Lemmas 4--6 were read for structure and not checked. Three filing observations are recorded in the contents below. Nothing here is independently reviewed.

Contents

Throughout, $f(z)=\prod_{\nu=1}^n(z-z_\nu)=z^n+\cdots$ and $E=\{|f(z)|\le1\}$, the closed set the paper calls the lemniscate domain (p. 97); the open set $\{|f(z)|<1\}$ is written $\mathring E$ where the paper distinguishes it (p. 102). Several site problems (#116, #511, #1038, #1039, #1048) are posed for the open set; where the transfer from the closed set matters, the result page says so as a filing observation.

• Introduction and § 1, the diameters of the components of $E$ (pp. 97--100). The paper "will give (at least partial) answers to some problems raised by Erdös, Herzog and Piranian [2]" and derive "some metric properties of continua of capacity 1" (p. 97). Every $E$ has logarithmic capacity 1 ([4]) and, quoted (p. 97), "the following approximation theorem holds [5]: Let $F$ be a closed bounded set with $\operatorname{cap}F=1$. Given any $\varepsilon>0$ and $\eta>0$, there exists a $\rho$ ($1<\rho<1+\eta$) and a polynomial $f(z)=z^n+\cdots$ such that the lemniscate $\{|f(z)|=\rho^n\}$ contains $F$ in its interior and is contained in an $\varepsilon$-neighborhood of $F$." Problem 8 of [2] asks whether $\sum\max(0,d_j-1)$ over the component diameters $d_j$ is bounded, and Problem 9 (revised form, [2, p. 148]) whether the number of components of diameter greater than a fixed $l>1$ is bounded. The paper states that both answers are negative, and that they stay negative when Problem 8 is asked with $d_j-l$, $l<4$, in place of $d_j-1$, and when Problem 9 is asked with any $l<4$ (p. 98). Theorem 1 (p. 98, quoted): "For each $0<l<4$ and $k=1,2,\cdots$, one can find a polynomial $f(z)=z^n+\cdots$ such that $E=\{|f(z)|\le1\}$ has at least $k$ different components of diameter greater than or equal to $l$." Proof: $F$ the union of the $k$ segments $[i\mu\delta,\,l+i\mu\delta]$, $\mu=1,\ldots,k$; a segment of length $l$ has capacity $l/4<1$, so $\operatorname{cap}F<1$ for small $\delta$, and the approximation theorem gives $f$ with $E\supset F$ inside a $\delta/3$-neighborhood of $F$. A filing observation, not a review verdict: the approximation theorem is quoted for $\operatorname{cap}F=1$ and a level $\rho^n>1$, and the proof applies it to a set of capacity below 1 at level 1 without spelling out the rescaling; the step was not reconstructed here. Theorem 2 (p. 98, quoted): "If $z^*\in E$ lies on a line of support of $E$, there exists a point $z\in E$ with $|z-z^*|>3\sqrt3/4\approx1.299$. The constant $3\sqrt3/4$ is best possible", answering Problem 10b of [2], which asked for such a $z$ with $|z-z^*|\ge2$; the paper cites its [9] for the failure of the constant $2$. Proof: an explicit map shows the half-disk of radius $3\sqrt3/4$ has capacity 1, and a lemniscate set inside it would equal it. Problem 7 of [2] (p. 98), quoted: "Let the zeros $z_\nu$ of $f(z)$ belong to the disk $|z|\le r$. Erdös, Herzog and Piranian [2, Problem 7] raised the question whether there is always a component of $E$ with diameter at least $2-r$ ($r<2$). The answer is negative for $r>1$." The example is $f(z)=z^n-r^n$, $r>1$: $E$ has $n$ components and, for $z$ in the component of the zero $r$, $|z-r|\le n^{-1}r^{-n+1}(1+O(n^{-1}))\to0$, so "the (common) diameter of the components of $E$ tends to $0$ as $n\to\infty$." Lemma 1 (pp. 98--99): with $z_0=\frac1n\sum z_\nu$ and $\sigma^2=\frac1n\sum|z_\nu|^2$, if $\sigma^2-|z_0|^2\le1$ the disk $|z-z_0|\le(1-\sigma^2+|z_0|^2)^{1/2}$ lies in $E$ (by the arithmetic-geometric mean inequality). Lemma 2 (p. 99): if a continuum $A\subset E$ contains all the zeros, $E$ is connected. Theorem 3 (p. 99, quoted): "Let $f(z)=\prod(z-z_\nu)$, $|z_\nu|\le r\le1$, and let $d_0$ be the diameter of the component $E_0$ of $E$ that contains $0$. Then $d_0\ge2$ for $0\le r\le1/2$, $d_0>1/r$ for $1/2<r\le(\sqrt5-1)/2$, $d_0>2-r^2$ for $(\sqrt5-1)/2\le r\le1$." Remarks (p. 99): $0\in E$ since $|f(0)|=\prod|z_\nu|\le1$, and the centroid $z_0$ lies in $E_0$; $d_0\ge2$ is sharp for $r\le1/2$ ($f=z^n$), and the 1958 polynomial $(z^n+1)(z-1)^2(z-e^{i\pi/n})^{-1}(z-e^{-i\pi/n})^{-1}$ has $d_0<1+\varepsilon$, so $d_0>1$ is best possible for $r=1$; Remark 3 observes that each of the three bounds $2$, $r^{-1}$ and $2-r^2$ is at least $2-r$, so Theorem 3 gives the affirmative answer to Problem 7 of [2] for $0<r\le1$. Proof (p. 100), four steps: $E_0$ meets $|z|>1$ unless $f=z^n$ (the polynomial $\prod(1-\bar z_\nu z)$ and the minimum principle); for $r\le1/2$, $E$ is connected and a continuum of capacity 1 has diameter at least 2; for the middle range, the disk of Lemma 1 and a case split on $|z_0|$ against $(1-2r^2)/(2r)$; for the last range, the point outside the unit disk and the disk of Lemma 1.
• § 2, the largest disk contained in $E$ (pp. 101--102). For zeros in a given compact set $A$, let $\rho$ be the radius of the largest disk contained in $E$ (p. 101). The paper recalls that $\rho$ has a positive lower bound $\rho_0=\rho_0(A)$ when $\operatorname{cap}A<1$ [2, Theorem 6], and that no lower bound independent of the degree $n$ exists when $A$ is a radius-$1$ disk or a length-$4$ segment, both of capacity $1$. Erdös, Herzog and Piranian asked whether $\rho\ge\mathrm{const}\cdot n^{-1}$ when $|z_\nu|\le1$ [2, Problem 3]; the paper calls Theorem 4 "a weaker estimate" and points also to [2, Problem 2]. Theorem 4 (p. 101, quoted): "If $|z_\nu|\le1$, the lemniscate domain $E$ contains a disk of radius $(2e)^{-1}n^{-2}$." Proof: $d_0>1$ by Theorem 3, $d_0\le4\operatorname{cap}E_0$ [6, p. 42], so $\operatorname{cap}E_0>1/4$; the derivative bound of [11], $|f'(z)|\le en^2/(2\operatorname{cap}E_0)<2en^2$ on $E_0$; integrating from a zero $z_\mu\in E_0$ to the nearest boundary point $z^*$ gives $1=|f(z^*)|<|z^*-z_\mu|\cdot2en^2$, so the disk $|z-z_\mu|\le1/(2en^2)$ lies in $E_0\subset E$. Lemma 3 (p. 101): for $\xi=z+z^{-1}$, $\xi'=z'+z'^{-1}$ on the upper unit semicircle, $|\xi-\xi'|\ge|z-z'|^2$. Theorem 5 (p. 101, quoted): "Let $-2\le z_\nu=\xi_\nu\le2$. Then the set $E\cap X$ contains a segment of length $1/8e^2n^4$ ($X$ denotes the real axis)", introduced with "I want to establish the conjecture of Erdös, Herzog and Piranian [2, p. 132] that $\rho\ge n^{-\gamma}$, where $\gamma$ denotes an absolute constant", and the introduction (p. 97) announces "In Section 2 it will be proved that $E$ contains a disk of radius const$\cdot n^{-4}$, if $z_\nu\in[-2,+2]$"; the theorem as printed gives a segment of $E\cap X$, not a disk. The conjecture printed on [2, p. 132] itself reads $|E\cap L|>n^{-c}$ for zeros on $[-2,2]$, which a real segment of length of order $n^{-4}$ in $E$ already gives. Proof (pp. 101--102): $g(z)=z^nf(z+z^{-1})$ of degree $2n$ with zeros on $|z|=1$; Theorem 4's proof gives a disk of radius $1/(2en^2)$ in $\{|g|\le1\}$ centered on $|z|=1$; an arc $B$ of the circle "of diameter $1/4en^2$" with $|g|\le1$ on $B$ and $\Im z\ge0$; $B^*=\{z+z^{-1}:z\in B\}$ "has length at least $(4en^2)^{-2}$, by Lemma 3", and $|f(\xi)|\le1$ on $B^*$. A filing observation, not a review verdict: the proof's last estimate as printed gives $(4en^2)^{-2}=1/(16e^2n^4)$, half the constant stated in the theorem; the discrepancy was not resolved here.
• § 3, upper bounds for geometric quantities associated with $E$ (pp. 102--105). Pólya [8]: the projection of a compact set of capacity 1 onto a line has linear measure $d\le4$. Theorem 6 (p. 102, quoted): "Let $f(z)=\prod_{\nu=1}^n(z-z_\nu)$, let $P$ be the projection of $E$ onto the real axis $X$, and let $d$ be the linear measure of $P$. Then $\operatorname{cap}P\le2^{-1/n}$, $d\le4\cdot2^{-1/n}$, with $\operatorname{cap}P=2^{-1/n}$ exactly if all $z_\nu$ lie on a parallel to $X$ and if $\mathring E=\{|f(z)|<1\}$ has $n$ components. (The result concerning $d$ was already known to P. Erdös and Bl. Sendov; see the remark after Problem 102, Wisk. Opgaven 20/3 (1957), p. 22.)" Remark: $d=4\cdot2^{-1/n}$ holds exactly when $P$ is one segment of capacity $2^{-1/n}$, which "can be shown" to happen if and only if $f(z)=T_n(2^{-1+1/n}z+c)$, $T_n$ the Chebyshev polynomial. Proof (pp. 102--103): $P\subset E^*\cap X$ for the polynomial $f^*$ with the real parts of the zeros, Fekete's theorem gives capacity $2^{-1/n}$ for $\{f^*\text{ real},\,|f^*|\le1\}$, and Pólya's $d\le4\operatorname{cap}P$. Then (p. 103): since $d\le4$ holds for every direction, the largest of the projection measures of $E$ is at most $4$. Writing $b$ for the smallest projection measure of $E$, the paper applies the approximation theorem to the "5-Stern" of [10, p. 73] and obtains a lemniscate domain with $b>2.386$, pointing to [2, Problem 10a], and then turns to an upper bound for $b$. Theorem 7 (p. 103, quoted): "Let $F$ be a closed bounded set with $\operatorname{cap}F=1$. Then the projection of $F$ onto a certain straight line has measure less than $3.30$." Proof: $F$ is enclosed by closed convex curves $L_\mu$ of total length below $10.36$ [12, Theorem 2]; with $b_\mu(\theta)$ the width of $L_\mu$ in direction $\theta$ and $\Lambda_\mu=\frac12\int_0^{2\pi}b_\mu(\theta)\,d\theta$ [1, p. 48], some $\theta_0$ has $\sum_\mu b_\mu(\theta_0)<10.36/\pi<3.30$, and the projection of $F$ onto a line perpendicular to the direction $\theta_0$ has measure at most that sum. Theorem 8 (p. 104): if $f\not\equiv z^n$ then $\operatorname{meas}[E\cap\{|z|=1\}]\le2\pi n/(n+1)$, by averaging $|f|^2$ over the $(n+1)$st roots of unity. Then (p. 104): "Problem 12a in [2] asks whether $\Lambda$ is greatest for $f(z)=z^n-1$. An affirmative answer would imply that $\Lambda\le2n+o(n)$." Theorem 9 (p. 104, quoted): "If $f(z)=z^n+\cdots$ and $\Lambda$ is the length of $C=\{|f(z)|=1\}$, then $\Lambda<74n^2$." Proof (pp. 104--105): $C$ is the real part of an algebraic curve of order $2n$, assumed by continuity to have only simple singularities and no real double points; at most $2n(2n-2)$ real inflection points [7] and, by Bézout, at most $2n(2n-1)$ points with horizontal tangent; fewer than $8n^2$ marked points cut $C$ into $m<8n^2$ arcs $C_k$ of constant curvature sign, each of capacity at most $\operatorname{cap}C=1$, whose convex hulls have perimeter below $9.2$ [10, Theorem 5]; so $\Lambda<9.2m<9.2\cdot8n^2<74n^2$.
• § 4, the connectedness of $E$ (pp. 105--110). Lemma 4 (p. 105): for $C(r)=\{|f(z)|=r^n\}$ and $\lambda(r)=\frac1{2\pi r^n}\int_{C(r)}|z|^2|f'(z)|\,|dz|$, $\lambda(r)>\sum|z_\nu|^2$ for $r>0$ ($\lambda$ is strictly increasing with limit $\sum|z_\nu|^2$ at $0$). Theorem 10 (pp. 106--107, quoted): "Let $z_0=\frac1n\sum_{\nu=1}^nz_\nu=0$ and $\sigma^2=\frac1n\sum_{\nu=1}^n|z_\nu|^2$. Then the following best possible results hold: (a) If $|z_\nu|\le\sqrt2/2$ or if $z_\nu\in[-1,+1]$, then $E$ is connected. (b) If $E$ is connected, then $|z_\nu|<2$ and $\sigma<\sqrt2$." Proof of (a): Lemma 1 puts the disk $|z|\le\sqrt2/2$ in $E$, or [2, Theorem 1] puts $[-1,0]$ and $[0,1]$ in $E$, and Lemma 2 applies; $(z^2-1/2)^m(z^2+a^2)$ with $a>\sqrt2/2$ and $z^2-a^2$ with $a>1$ show sharpness. Proof of (b) (p. 107): with $z_0=0$, $w=f(z)^{1/n}=z+a_2^*z^{-1}+\cdots$ is univalent outside $E$, its inverse $\phi(w)=w+\sum b_\mu w^{-\mu}$ is univalent in $|w|>1$, so $E$ lies in the closed disk $|z|\le2$ [6, p. 42], and each zero, an interior point of $E$, has $|z_\nu|<2$; Lemma 4 and the area theorem $\sum\mu|b_\mu|^2\le1$ give $\sigma^2<2$. The polynomial $T_n(2^{1/n-1}z)$, whose $E$ is connected, shows the bounds in (b) cannot be improved (pp. 107--108). Lemma 5 (p. 108): $f(z)^{1/n}$ is univalent outside the convex hull of the zeros. Theorem 11 (p. 108): if $A$ is a closed bounded convex set of capacity $\kappa\le1/2$ with conformal center $0$, the zeros lie in $A$ and their centroid is $0$, then $E$ is connected and contains $A$. Then (p. 109), for a continuum $F$ of capacity 1 with width $b$ and diameter $d$, the paper recalls Problem 15 of [2] (bounds for $b$, $d$, $bd$), its own $b<2.920$ [10, Theorem 6], and completes the proof of the inequality asserted in [9], whose proof "was not correctly formulated, as Prof. Herzog kindly pointed out": the sentence prints the assertion as $b^2+d^2\le63/3$, and the completed proof derives $b^2+d^2\le64/3$, "which is the asserted inequality"; "Probably $b^2+d^2\le16$ holds (with equality for a segment of length 4)", proved in [10, Theorem 9] when $F$ is convex or contains a segment of length $d$. Theorem 12 (p. 109): for a continuum $F$ of capacity 1 symmetric about $0$, either (I) $b\le2\sqrt2$, $b^2+d^2\le16$ and $bd\le8$, or (II) $b\le2$, $b^2+d^2\le18$ and $bd\le4\sqrt3$; remarks give a symmetric continuum with $b>2.18$ and one with $bd>6.15$ [9], and the sharp $b^*\le2\sqrt2$, $b^*d\le8$ for the width $b^*$ measured parallel to a diameter (equality for two perpendicular segments of length $2\sqrt2$).
• § 5, convexity (pp. 110--115). Erdős, Herzog and Piranian [2, Theorem 11] proved $E$ convex when $|z_\nu|\le\sin(\pi/8)/(1+\sin(\pi/8))\approx0.277$. Theorem 13 (p. 110, quoted): "If one of the conditions (a) $|z_\nu|\le0.320$, (b) $|z_\nu|\le0.424$ and $z_0=0$ is satisfied, then $E$ is convex." Proof (pp. 110--111) by the area theorem and a convexity criterion for the level curve of the exterior map (Hilfssatz 4b of [13]). Then (pp. 111--112), for $f(z)=\prod_{k=1}^m(z-z_k)^{p_k}$ with distinct $z_k$ and positive integer exponents $p_k$, and $E=\{|f(z)|\le1\}$ with the maximal number $m$ of components, the paper recalls that "H. Grunsky (see [2, Problem 16]) raised the question whether all components must be convex" and announces a counterexample. Theorem 14 (p. 112, quoted): "Let $f(z)=z^p(z-a)$. If $a-(1+p^{-1})\cdot p^{1/(p+1)}$ is positive and sufficiently small, and $p$ is sufficiently large, then the set $E=\{|f(z)|\le1\}$ has two components, one of which is not convex." Proof: with $\xi=p^{1/(p+1)}$ and $f_p(z)=z^p(z-(1+p^{-1})\xi)$, $f_p(\xi)=-1$ and $f_p'(\xi)=0$, so $\xi$ is a double point of $\{|f_p|=1\}$ whose two branches have tangents $y=\pm(x-\xi)$, and near $\xi$ the set $E_p=\{|f_p|\le1\}$ lies in $S=\{x+iy:|y|\le1.1|\xi-x|\}$; the point $z^*=0.5+0.6i$ has $f_p(z^*)\to0$, so $z^*\in E_p$ for large $p$, while $z^*\notin S$ since $\xi\to1$; the segment from $z^*$ to $\xi$ leaves $E_p$, and increasing $a$ slightly separates the two components and keeps the component of $0$ nonconvex. Then (p. 112) the paper turns to Problem 13 of Erdös, Herzog and Piranian [2], quoted: "Let $z_\nu$ be $n$ complex numbers which satisfy $|z_\mu-z_\nu|\le2$ ($\mu,\nu=1,\cdots,n$). Is $\prod_{\nu=1}\prod_{\mu\ne\nu}|z_\mu-z_\nu|$ maximal if the $z_\nu$ are the vertices of a regular $n$-gon of diameter 2?" It writes $\Delta_n$ for the maximum, display (10) $\Delta_n=\max_{|z_\mu-z_\nu|\le2}\prod_{\nu=1}^n\prod_{\mu\ne\nu}|z_\mu-z_\nu|$, and notes that the conjecture would give $\Delta_n=n^n$ for even $n$ and $\Delta_n=n^n(\cos\pi/2n)^{-n(n-1)}$ for odd $n$, the latter equal to $n^n(1-\pi^2/8n^2+\cdots)^{-n(n-1)}\sim n^ne^{\pi^2/8}$. Lemma 6 (pp. 112--113): for a convex continuum $K$ of capacity 1 there are monic polynomials $f_n$ of every degree with zeros in $K$ and $\max_K|f_n|\le4$ (the zeros are the images of the $n$th roots of unity under the exterior map, and a starlike univalent function bounds the product). Theorem 15 (p. 113, quoted): "Let $K$ be a convex continuum of capacity 1. Then, for $z_\nu\in K$ ($\nu=1,\cdots,n$), $\prod_{\nu=1}^n\prod_{\mu\ne\nu}|z_\mu-z_\nu|\le2^{4(n-1)}n^n$." Proof (pp. 113--114): the product is the squared Vandermonde determinant, whose columns may be replaced by $1,f_1(z_j),\ldots,f_{n-1}(z_j)$; Hadamard's determinant theorem gives $n^n\prod_k\max_K|f_k|^2$ (the paper attributes this inequality to Szegö, citing footnote 7 on p. 236 of [3]), and Lemma 6 bounds each factor by 16. Theorem 16 (p. 114, quoted): "If $\Delta_n$ is defined by (10), then $\Delta_n\le2^{4(n-1)}\cdot n^n$." Proof: for $K$ the convex hull of a maximal system, Theorem 15 scaled gives (12) $\Delta_n\le2^{4(n-1)}n^n(\operatorname{cap}K)^{n(n-1)}$, and $2\operatorname{cap}K\le\operatorname{diam}K\le2$. Remark (pp. 114--115), "in favor of the conjecture", quoted: "The convex hull $K_n$ of a maximal system $\{z_n^{(n)},\cdots,z_n^{(n)}\}$ [sic] is nearly a disk, for large $n$" (Remark, p. 114), since a subsequence of hulls converging to a non-disk convex set $K_0$ of diameter at most 2 would have $\operatorname{cap}K_{n_k}\le1-\delta<1$, and (12) would give $\Delta_{n_k}<1$, against $\Delta_{n_k}\ge n_k^{n_k}$.

Compiled scope

The paper is compiled at statement depth for the results the citing problems consume, each with a result page: Theorem 1 (p. 98), the $z^n-r^n$ example (p. 98), Theorem 3 (p. 99), Theorem 4 (p. 101), the projection example with Theorem 7 (p. 103), Theorem 9 (p. 104), Theorem 10 (pp. 106--107), Theorem 14 (p. 112) and Theorems 15--16 (pp. 113--114). Theorems 2, 5, 6, 8, 11, 12 and 13 and Lemmas 1--6 are recorded above as statements read on the page images. The short proofs named in the read status were followed; no proof was checked line by line, and nothing here is independently reviewed.

Bears on. #1043: the passage on p. 103, "By applying the approximation theorem to the "5-Stern" [10, p. 73] we obtain a lemniscate domain with $b>2.386$ (compare [2, Problem 10a])", where $b$ is "the minimum of the measures of the projections of $E$", is the negative answer to the problem's question (a line onto which $\{|f|\le1\}$ projects to measure at most 2): for that polynomial every projection has measure above $2.386$. The construction rests on the approximation theorem and on the 5-Stern of the author's [10, p. 73], not held, and not on the 1959 note [9], which the paper cites only for Problem 10b (Theorem 2, p. 98) and for the width and diameter bounds of p. 109. Theorem 7 (p. 103) bounds the other side, some projection of any capacity-1 set has measure below $3.30$, and Theorem 6 (p. 102) gives $d\le4\cdot2^{-1/n}$ for the projection onto any fixed line. #1045: display (10) on p. 112 is the problem's ordered product $\prod_{i\ne j}|z_i-z_j|$ under the same diameter constraint, and Theorem 16 (p. 114), "$\Delta_n\le2^{4(n-1)}\cdot n^n$", is a general upper bound, from Theorem 15 (p. 113) for points in a convex continuum of capacity 1; the remark on pp. 114--115 shows that for large $n$ the convex hull of any maximizing configuration is close to a disk. The paper does not determine $\Delta_n$ or decide the regular-polygon question; the values it records for the conjecture are $n^n$ for even $n$ and $n^n(\cos\pi/2n)^{-n(n-1)}$ for odd $n$, the even value later refuted for every even $n\ge4$ by Danzer and Pommerenke (1967). #1047: Theorem 14 (p. 112), quoted above, the set $E=\{|f(z)|\le1\}$ of $f(z)=z^p(z-a)$ with two components, one of which is not convex, answers Grunsky's question, the problem's, in the negative with $m=2$ distinct roots at $c=1$ for the closed sublevel set the problem uses; the counterexample of high degree with a multiple root that Goodman (1966) attributes to the author, before his quartic with simple roots. #1048: the example on p. 98, $f(z)=z^n-r^n$ with $1<r<2$, whose $n$ components have common diameter tending to $0$, is the negative answer for $1<r<2$ ("The answer is negative for $r>1$"), and Theorem 3 (p. 99) with its Remark 3 is the affirmative answer for $0<r\le1$ for the closed set, which carries over to the open set for $0<r\le1/2$ and is left undecided by the theorem for $1/2<r\le1$; the problem is posed for the open set $\{|f|<1\}$ and a strict inequality, and the example transfers since the open set's components lie inside the closed set's. #114: Theorem 9 (p. 104), "$\Lambda<74n^2$" for the length of $\{|f(z)|=1\}$, is an upper bound only; the paper records that an affirmative answer to Problem 12a of [2], the problem's question, "would imply that $\Lambda\le2n+o(n)$", and decides nothing about it. #116: Theorem 4 (p. 101), a disk of radius $(2e)^{-1}n^{-2}$ inside $E$ when $|z_\nu|\le1$, gives $|\{|f|<1\}|\ge\pi(2e)^{-2}n^{-4}$, the polynomial lower bound the problem asks for (the open disk of that radius lies in the interior of $E$, which is the open set); the paper says nothing about a $(\log n)^{-O(1)}$ bound. #509: the paper states no result on covering $E$ by disks with radii summing to at most 2. Its connected case is the proof of Theorem 10(b) (p. 107): for a connected $E$ with the centroid of the zeros at $0$, "$E$ is contained in $|z|\le2$", one disk of radius 2 about the centroid; Theorem 2 (p. 98) and Theorem 6 (p. 102) bound related quantities. #511: Theorem 1 (p. 98), at least $k$ components of diameter at least $l$ for every $l<4$ and every $k$, is the negative answer to the problem's question (boundedly many components of diameter above a fixed $c>1$, independent of the degree), stated by the paper as the negative answer to Problems 8 and 9 of [2]; the problem is posed for the open set, and the approximation theorem puts the segments in the interior of $E$, which is that open set. The 2025 rediscovery is filed as huang_2025_many_lemniscates_large_diameter. #1038: the paper states no result on the infimum or the supremum of $|\{x:|f(x)|<1\}|$ for real zeros in $[-1,1]$; Theorem 5 (p. 101) gives a real segment of length $1/8e^2n^4$ in $E$ for real zeros in the wider interval $[-2,2]$, with the printed-constant observation above, and the proof of Theorem 10(a) (p. 107) records that for zeros in $[-1,1]$ with centroid $0$ both $[-1,0]$ and $[0,1]$ lie in $E$ by [2, Theorem 1]. #1039: Theorem 4 (p. 101) is the bound $\rho\ge(2e)^{-1}n^{-2}$ on the problem's $\rho(f)$, introduced (p. 101) as "a weaker estimate" than the $\rho\ge\mathrm{const}\cdot n^{-1}$ of [2, Problem 3], the problem's question, which the paper leaves open.

Results.

• Theorem 1 (p. 98): for each $0<l<4$ and each $k$, a monic polynomial whose $E$ has at least $k$ components of diameter at least $l$.
• Example, p. 98: $f(z)=z^n-r^n$, $1<r<2$: $n$ components of common diameter tending to $0$; no component of diameter at least $2-r$ for large $n$.
• Theorem 3 (p. 99): for $|z_\nu|\le r\le1$ the component of $0$ has diameter $\ge2$, $>1/r$ or $>2-r^2$ in three ranges, in every case at least $2-r$.
• Theorem 4 (p. 101): for $|z_\nu|\le1$, $E$ contains a disk of radius $(2e)^{-1}n^{-2}$.
• Example, p. 103: a lemniscate set with every projection of measure above $2.386$; with Theorem 7, some projection of any capacity-1 set has measure below $3.30$.
• Theorem 9 (p. 104): the length of $\{|f(z)|=1\}$ is less than $74n^2$.
• Theorem 10 (pp. 106--107): with centroid $0$, $E$ is connected for zeros in $|z|\le\sqrt2/2$ or in $[-1,1]$; a connected $E$ lies in $|z|\le2$, with $|z_\nu|<2$ and $\sigma<\sqrt2$.
• Theorem 14 (p. 112): $z^p(z-a)$ with two components, one not convex.
• Theorem 16 (p. 114): $\Delta_n\le2^{4(n-1)}n^n$, from Theorem 15 (p. 113) for points in a convex continuum of capacity 1.

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