Erdos 1958 metric properties polynomials

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problem_1: Asks for the supremum and infimum of the measure of the real part of the set where |f| < 1 when all zeros lie in [-r,r], and conjectures the upper bound 2 sqrt 2 when they lie in [-1,1]; the source of Problem 1038.

problem_10: Asks whether, for every monic polynomial, the closure of the set where |f| < 1 has a projection of measure at most 2 onto some line, and whether a point of the closure on a support line is at distance at least 2 from another point of it; the source of Problem 1043.

problem_11: Asks for the infimum, over monic polynomials with all zeros in the open unit disk, of the largest boundary length among the components of the set where |f| < 1; the source of Problem 1044.

problem_12: Asks whether, in fixed degree, the lemniscate |f(z)| = 1 is longest for z^n - 1, whether it has length at least 2 pi when E is connected, and what its infimum length is for zeros in the open unit disk; the first question is Problem 114.

problem_13: For fixed n, asks for the maximum of the modulus of the discriminant of a monic polynomial whose zeros are pairwise at distance at most 2, and whether it is attained at a regular n-gon whose greatest diagonal has length 2; the source of Problem 1045.

problem_14: For a K-polynomial, one whose set |f| < 1 is connected, asks whether that set lies in a disk of radius 2 and whether the disk can be centered at the centroid of the zeros; the source of Problem 1046.

problem_16: Grunsky's question: for a polynomial with m distinct zeros of any positive integer multiplicities and a level c small enough that the lemniscate |f| = c consists of m distinct loops, are all the loops convex? The source of Problem 1047.

problem_2: Asks for the polynomials with zeros in the closed unit disk minimizing the area of the set where |f| < 1 in each degree, and for estimates of that least area alpha_n, for instance whether alpha_n > n^{-c}; the source of Problem 116.

problem_3: For zeros in the closed unit disk, asks for the asymptotic behavior of the radius rho_n of the largest disk that the set where |f| < 1 must contain, and whether rho_n > c/n; z^n - 1 shows c <= pi/2. The source of Problem 1039.

problem_4: For a closed infinite set F, asks whether the infimum mu(F) of the area of the set where |f| < 1, over polynomials with all zeros in F, is determined by the transfinite diameter of F, and in particular whether it is 0 when that diameter is at least 1; the source of Problem 1040.

problem_5: For a monic polynomial with all zeros in the open unit disk, asks whether some path of length less than 2 inside the set where |f| < 1 joins two of the zeros; the source of Problem 1041.

problem_6: Asks whether E can have n components when the zeros lie in a closed set of transfinite diameter 1 not inside any closed disk of radius 1, and whether a transfinite diameter below 1 caps the count at (1-c)n for large n; the source of Problem 1042.

problem_7: For a monic polynomial with all zeros in the open disk of radius r < 2, asks whether the set where |f| < 1 has a component of diameter greater than 2 - r; the source of Problem 1048.

problem_9: Asks whether N_n, the most components of diameter greater than 1 that the closure of E can have for degree n with zeros in D, is bounded; a note added in proof answers no, with N_n >= n/2, by perturbing z^n + 1. The restricted precursor of Problem 511.

problem_p148: With no restriction on the zeros, asks whether N_n(c), the supremum of the number of components of the set where |f| < 1 with diameter greater than 1 + c, is bounded in n for each fixed c > 0; the source of Problem 511.

theorem_1: For a monic polynomial with all zeros in [-1,1] and centroid in [0,1], the real part of the set where |f| < 1 contains an interval J containing (0,1), holding at least n/2 of the zeros and of length at least sqrt 2, while the set misses (-infinity, -sqrt 2].

theorem_10: If the set where |f| < 1 is connected and n > 1, the discriminant of the monic polynomial has modulus below n^n; if only its closure is connected, the modulus is at most n^n, with equality exactly when |f| = 1 at every zero of f'. Netanyahu's conjecture, also proved by W. H. Fuchs.

theorem_11: If all zeros of the monic polynomial lie in a disk of radius r_0 = sin(pi/8)/(1 + sin(pi/8)), the set where |f| < 1 is convex; the example (z-r)^m(z+r) shows r_0 cannot be replaced by any constant above 1/2.

theorem_12: For monic polynomials with zeros in the closed unit disk and maximum modulus on the unit circle greater than (1+c)^n, there are c_1(c) < 1 and c_2(c) > 0 with |f| < c_1^n on a subset of the open disk of measure at least c_2.

theorem_2: The supremum delta(r) of the diameter of the real part of the set where |f| < 1, over monic polynomials with all zeros in [-r,r], is 2 sqrt(1+r^2) for 0 <= r <= 3/4 and 1+2r for r >= 3/4.

theorem_3: If every zero of the monic polynomial is -1 or 1, the measure of the real part of the set where |f| < 1 is at most 2 sqrt 2; the paper offers it as the evidence for its conjecture that the same bound holds for all zeros in [-1,1].

theorem_4: For a monic polynomial with all zeros in the closed unit disk, the area of the set where |f| < 1 is at most 4 (pi times the area of its part in the open disk)^{1/2}; with MacLane's theorem this gives infimum 0 for that area over polynomials with all zeros on the unit circle.

theorem_5: For a monic polynomial with all zeros on the unit circle, the arc length of the part of the circle where |f| < 1 lies strictly between 0 and 2 pi, and neither constant can be improved.

theorem_6: If a closed set F has transfinite diameter less than 1, there is rho(F) > 0 such that, for every monic polynomial with all zeros in F, the set where |f| < 1 contains a disk of radius rho(F).

theorem_7: For all sufficiently large n, the polynomial obtained from z^n + 1 by moving its two zeros nearest 1 to a double zero at 1 has a closed set where |f| <= 1 with exactly n - 1 components, so Szegő's bound n - 1 for zeros in the closed unit disk cannot be lowered for large n.

theorem_8: For a monic polynomial with real zeros, not all equal, and a > 0 with |f(a)| = |f(-a)|, the modulus on the circle of radius a exceeds |f(a)| off the real axis; consequently, for real zeros, the diameters of the components of the set where |f| < 1 add up to the measure of its real part.

theorem_9: S(r), the supremum over all degrees of the largest sum of the diameters of the components of the set where |f| < 1 for zeros in the closed disk of radius r, equals 2 sqrt(1+r^2) for 0 <= r <= 1/2, and S(1-b) > (1/2 - eps)(1 - 1/e) log(1/b) for every eps > 0 once b is small.

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P. Erdős, F. Herzog, G. Piranian: Metric properties of polynomials, J. Analyse Math. 6 (1958), 125--148 (MR 21 #123; Zentralblatt 88,253); DOI 10.1007/bf02790232.

For a monic polynomial f(z) = prod (z - z_v) the paper studies the set E = E(f) where |f(z)| < 1. Theorem 1 shows that if the zeros lie in I = [-1,1] with centroid in [0,1], then the part of E on the real axis L contains an interval J containing (0,1), containing at least n/2 of the zeros, with |J| >= sqrt 2, while E misses (-infinity, -sqrt 2]. Theorem 4 gives |E| <= 4 (pi |E intersect D|)^{1/2} for zeros in the closed unit disk, whose corollary (with MacLane's theorem) is that inf |E| = 0 for zeros on the unit circle; Theorem 5 shows 0 < |E intersect C| < 2 pi for zeros on the unit circle C, with both constants best possible, and Theorem 6 that if a closed set F has transfinite diameter less than 1 then E(f) contains a disk of radius rho(F) whenever all zeros lie in F. Theorem 7 exhibits, for every sufficiently large n, a polynomial with all zeros on the unit circle whose closure of E has exactly n-1 components, so Szegő's bound n-1 for zeros in the closed unit disk cannot be lowered for those n; Theorem 9 concerns S(r), the supremum over all degrees of the largest sum of the diameters of the components of E when the zeros lie in the closed disk of radius r (S(r) = 2 sqrt(1+r^2) for 0 <= r <= 1/2, and S(1-b) > (1/2 - eps)(1 - e^{-1}) log(1/b) for every eps > 0 once b is small enough); Theorem 10 proves the conjecture raised by E. Netanyahu, and proved independently by W. H. Fuchs, that |discriminant(f)| < n^n for K-polynomials of degree n > 1 and <= n^n for closed-K-polynomials, with equality characterized; Theorem 11 shows E is convex when the zeros lie in a disk of radius sin(pi/8)/(1 + sin(pi/8)); Theorem 12 shows that if the zeros lie in the closed unit disk and the maximum modulus on C exceeds (1+c)^n, then |f| < c_1^n on a subset of the open disk of measure at least c_2, for constants c_1(c) < 1 and c_2(c) > 0. Theorem 2 gives the supremum delta(r) of diam(E intersect L) for zeros in [-r,r], Theorem 3 the bound 2 sqrt 2 for |E intersect L| when every zero is -1 or 1, and the Corollary to Theorem 8 that for real zeros the diameters of the components of E sum to |E intersect L|. The paper is the source of the sixteen numbered Problems 1--16 on the length of E intersect L, the least measure alpha_n, the inradius rho_n, the role of transfinite diameter, component diameters and counts, projections, lemniscate lengths, discriminant maxima and loop convexity; an added-in-proof note settles Problem 9 negatively by showing N_n >= n/2. The listed problems 114, 116, 511 and 1038--1048 draw their statements from this cluster of theorems and open problems on lemniscates.

The copy read for this card is the Rényi Institute's Erdős archive scan (24 pages, printed pp. 125--148 = PDF pp. 1--24). Read status: claims checked. The statements of Theorems 1--12 with their corollaries, the definitions of Section 7, the Remark after Theorem 10, Problems 1--7, 9--14 and 16, and the note added in proof (p. 148) were read clause by clause on the page images on 2026-10-08; Problems 8 and 15 were not checked. The proofs were read for structure, and the identity behind Theorem 10 was followed; no proof was independently verified. The scan carries no notice; the publisher's article page shows "© Hebrew University of Jerusalem" under "Reprints and permissions" as subscription content with no open access designation (https://link.springer.com/article/10.1007/BF02790232), every other right reserved.

The discriminant question of Problem 1045. Theorem 10 (p. 143) proves the conjecture raised by E. Netanyahu, also proved independently by W. H. Fuchs: if f is a K-polynomial, meaning that E(f) is connected, then |D(f)| < n^n for n > 1, and if f is a closed-K-polynomial, meaning that the closure of E(f) is connected, then |D(f)| <= n^n, with equality exactly when |f| = 1 at every zero of f'. The mechanism is the unnumbered identity immediately below Theorem 10: writing f'(z) = n prod_{v=1}^{n-1} (z - z'v), |D(f)| = prod{v<mu} |z_v - z_mu|^2 = prod_mu |f'(z_mu)| = n^n prod_{v=1}^{n-1} |f(z'v)|; connectedness of E(f) is characterized just before the theorem by |f(z'v)| < 1 for every critical point, connectedness of the closure by <= 1, and substitution gives the strict and weak bounds and the equality criterion. Problem 13, after a one-sentence remark on the same page, asks for the maximum of |D(f)| when |z_mu - z_v| <= 2 for all mu < v, and whether it is attained when the z_v are the vertices of a regular n-gon whose greatest diagonal has length 2. This is exactly Problem 1045: its ordered product prod{i != j} |z_i - z_j| equals prod{i<j} |z_i - z_j|^2 = |D(f)|, so the catalog and the source use the same objective, not merely objectives with the same maximizers. Theorem 10 does not answer Problem 13: its hypothesis controls the critical values through a connected lemniscate, whereas Problem 13 assumes only a diameter bound on the roots, and the paper supplies no implication from the latter to the former. Problem 13 is posed without a proof of regular-polygon optimality, a general upper bound for the diameter-constrained class, or a classification of the extremizers.

The regular-polygon sentence on p. 143 is posed as a question; it was later cited as the three authors' conjecture and refuted for even orders k >= 4 by Danzer and Pommerenke (1967), who define D_k in their equation (1.1) as the maximum of the same ordered distance product, cite Problem 13 and p. 143, and state just before their Theorem 1 that the conjecture is false for even k: an optimal diameter-2 set must have a connected diameter graph (1967, pp. 101--102), while the diameter graph of a regular even k = 2l polygon is a disjoint union of l antipodal edges (p. 103), so that polygon is not optimal once l >= 2; perturbing alternating radii gives a strictly larger product for k >= 6 (their (1.3) and (1.4)), and for k = 4 their exact value D_4, about 294.08 (their (1.7)), exceeds 4^4. This is a later partial negative answer to the 1958 question, not a correction or result contained in the 1958 article; it covers every even order k >= 4 (for k = 2 two points at distance 2 are optimal, D_2 = 4 in their Theorem 1) and leaves the odd-order case outside it.

Source: https://users.renyi.hu/~p_erdos/1958-05.pdf.

Results. Theorems: Theorem 1 (p. 126), Theorem 2 (p. 128), Theorem 3 (p. 131), Theorem 4 and Corollary (p. 133), Theorem 5 (p. 134), Theorem 6 (p. 135), Theorem 7 (p. 136), Theorem 8 and Corollary (pp. 139--140), Theorem 9 (p. 140), Theorem 10 and Remark (p. 143), Theorem 11 (p. 143), Theorem 12 (p. 145). Problems: Problem 1 (p. 131), Problem 2 (pp. 133--134), Problem 3 (p. 134), Problem 4 (pp. 135--136), Problem 5 (p. 139), Problem 6 (p. 139), Problem 7 (p. 142), Problem 9 and its answer added in proof (pp. 142, 148), Problem 10 (p. 142), Problem 11 (p. 142), Problem 12 (p. 142), Problem 13 (p. 143), Problem 14 (p. 143), Problem 16 (p. 145) and the question added in proof (p. 148). Problems 8 and 15, which no listed problem draws on, have no page.

Bears on. Each problem below takes its question from the paper; the paper poses these questions and, apart from the note on Problem 9, answers none of them.

• #114: the first question of Problem 12 (p. 142), in the same terms.
• #116: the second question of Problem 2 (pp. 133--134), whether $\alpha_n>n^{-c}$; the Corollary to Theorem 4 gives $\inf_n\alpha_n=0$ (a step drawn on its page). The alternative bound $(\log n)^{-O(1)}$ in the problem's statement is not in the paper.
• #511: the question added in proof (p. 148), with the threshold $c>1$ for the paper's $1+c$, $c>0$; the note's answer to Problem 9 concerns the threshold $1$ only.
• #1038: Problem 1 (p. 131) for $r=1$, with the paper's conjecture $2\sqrt2$ for the supremum (Theorem 3 is its zeros-at-$\pm1$ case) and its remark that the infimum is below $2$ (p. 132); Theorems 1 and 2 give the bounds $\sqrt2$ and $3$ by steps drawn on their pages.
• #1039: Problem 3 (p. 134), the asymptotic behavior of the inradius $\rho_n$ and whether $\rho_n>c/n$.
• #1040: Problem 4 (pp. 135--136), in the same terms; Theorem 6 gives $\mu(F)>0$ below transfinite diameter $1$ (a step drawn on its page), and the paper says that the segment and disk cases follow from Chebyshev polynomials with Theorem 8 and from Theorem 4, respectively.
• #1041: Problem 5 (p. 139), in the same terms.
• #1042: Problem 6 (p. 139); the paper's second question carries the qualifier "when $n$ is large", which the problem's statement omits.
• #1043: the first question of Problem 10 (p. 142), with $\{|f|\le1\}$ for the paper's $\bar E$, the same set.
• #1044: Problem 11 (p. 142), with zeros in the closed disk where the paper prints the open disk $D$.
• #1045: Problem 13 (p. 143): the diameter-constrained discriminant maximum and the regular-polygon question, the problem's exact objective; Theorem 10 bounds a different class.
• #1046: the first question of Problem 14 (p. 143), without its centroid clause.
• #1047: Problem 16 (p. 145), Grunsky's question, stated for the components of $\{|f|\le c\}$ where the paper asks about the loops of $|f|=c$.
• #1048: Problem 7 (p. 142), with zeros in the closed disk $|z|\le r$ where the paper prints the open disk $D_r$.

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