A problem of Erdős, Herzog, and Piranian [EHP58]. It is also listed as Problem 4.10 in [Ha74], where it is attributed to Erdős.

Let the maximal length of such a curve be denoted by $f(n)$.

• The length of the curve when $p(z)=z^n-1$ is $2n+O(1)$, and hence the conjecture implies in particular that $f(n)=2n+O(1)$.


• Dolzhenko [Do61] proved $f(n) \leq 4\pi n$, but few were aware of this work.


• Pommerenke [Po61] proved $f(n)\ll n^2$.


• Borwein [Bo95] proved $f(n)\ll n$ (Borwein was unaware of Dolzhenko's earlier work). The prize of \$250 is reported by Borwein [Bo95].


• Eremenko and Hayman [ErHa99] proved the full conjecture when $n=2$, and $f(n)\leq 9.173n$ for all $n$.


• Danchenko [Da07] proved $f(n)\leq 2\pi n$.


• Fryntov and Nazarov [FrNa09] proved that $z^n-1$ is a local maximiser, and solved this problem asymptotically, proving that\[f(n)\leq 2n+O(n^{7/8}).\]


• Tao [Ta25] has proved that $p(z)=z^n-1$ is the unique (up to rotation and translation) maximiser for all sufficiently large $n$.
  

Erdős, Herzog, and Piranian [EHP58] also ask whether the length is at least $2\pi$ if $\{ z: \lvert f(z)\rvert<1\}$ is connected (which $z^n$ shows is the best possible). This was proved by Pommerenke [Po59].

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The lower bound is easy: this is $\geq n$ and equality holds if and only if $p(z)=z^n$. The assumption that the set is connected is necessary, as witnessed for example by $p(z)=z^2+10z+1$.

The Chebyshev polynomials show that $n^2/2$ is best possible here. Erdős originally conjectured this without the $o(1)$ term but Szabados observed that was too strong. Pommerenke [Po59a] proved an upper bound of $\frac{e}{2}n^2$.

Eremenko and Lempert [ErLe94] have shown this is true, and in fact Chebyshev polynomials are the extreme examples.

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Conjectured by Erdős, Herzog, and Piranian [EHP58]. The lower bound $\gg n^{-4}$ follows from a result of Pommerenke [Po61]. The lower bound $\gg (\log n)^{-1}$ was proved by Krishnapur, Lundberg, and Ramachandran [KLR25].

Wagner [Wa88] proves, for $n\geq 3$, the existence of such polynomials with\[\lvert\{ z: \lvert p(z)\rvert <1\}\rvert \ll_\epsilon (\log\log n)^{-1/2+\epsilon}\]for all $\epsilon>0$. Krishnapur, Lundberg, and Ramachandran [KLR25] improved this upper bound to $\ll (\log\log n)^{-1}$.

In [EHP58] they also ask to determine the polynomials which achieve the minimum possible value of this measure.

Pólya [Po28] showed the upper bound\[\lvert\{ z: \lvert p(z)\rvert <1\}\rvert \leq \pi\]always holds, and this is achieved only when the $z_i$ are identical.

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This is Problem 4.1 in [Ha74] where it is attributed to Erdős.

The weaker conjecture that $\limsup M_n=\infty$ was proved by Wagner [Wa80], who show that there is some $c>0$ with $M_n>(\log n)^c$ infinitely often.

The second question was answered by Beck [Be91], who proved that there exists some $c>0$ such that\[\max_{n\leq N} M_n > N^c.\]Erdős (e.g. see [Ha74]) gave a construction of a sequence with $M_n\leq n+1$ for all $n$. Linden [Li77] improved this to give a sequence with $M_n\ll n^{1-c}$ for some $c>0$.

The third question seems to remain open.

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Originally a conjecture of Littlewood. The answer is yes (for all $n\geq 2$), proved by Balister, Bollobás, Morris, Sahasrabudhe, and Tiba [BBMST20].

In Problem 4.31 of [Ha74] Erdős asks whether there exists a constant $c>0$ such that $\max_{\lvert z\rvert=1}\lvert P(z)\rvert > (1+c)\sqrt{n}$ for all large $n$.

See also [230].

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This is Problem 4.31 in [Ha74], in which it is described as a conjecture of Erdős and Newman.

The lower bound of $\sqrt{n}$ is trivial from Parseval's theorem. The answer is no (contrary to Erdős' initial guess). Kahane [Ka80] constructed 'ultraflat' polynomials $P(z)=\sum a_kz^k$ with $\lvert a_k\rvert=1$ such that\[P(z)=(1+o(1))\sqrt{n}\]uniformly for all $z\in\mathbb{C}$ with $\lvert z\rvert=1$, where the $o(1)$ term $\to 0$ as $n\to \infty$.

For more details see the paper [BoBo09] of Bombieri and Bourgain and where Kahane's construction is improved to yield such a polynomial with\[P(z)=\sqrt{n}+O(n^{\frac{7}{18}}(\log n)^{O(1)})\]for all $z\in\mathbb{C}$ with $\lvert z\rvert=1$.


See also [228].

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This is Problem 4.4 in [Ha74], where it is attributed to Erdős.

First investigated by Rényi and Rédei [Re47]. Erdős [Er49b] proved that $f(k)<k^{1-c}$ for some $c>0$. The conjecture that $f(k)\to \infty$ is due to Erdős and Rényi.

This was solved by Schinzel [Sc87], who proved that\[f(k) > \frac{\log\log k}{\log 2}.\]In fact Schinzel proves lower bounds for the corresponding problem with $P(x)^n$ for any integer $n\geq 1$, where the coefficients of the polynomial can be from any field with zero or sufficiently large positive characteristic.

Schinzel and Zannier [ScZa09] have improved this to\[f(k) \gg \log k.\]

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Cartan proved this is true with $2$ replaced by $2e$, which was improved to $2.59$ by Pommerenke [Po61]. Pommerenke [Po59] proved that $2$ is achievable if the set is connected (see [1046]).

The generalisation of this to higher dimensions was asked by Erdős as Problem 4.23 in [Ha74].

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Erdős and Offord [EO56] showed that the number of real roots of a random degree $n$ polynomial with $\pm 1$ coefficients is $(\frac{2}{\pi}+o(1))\log n$.

It is ambiguous in [Er61] whether Erdős intended the coefficients to be uniformly chosen from $\{-1,1\}$ or $\{0,1\}$. In the latter case, the constant $\frac{2}{\pi}$ should be $\frac{1}{\pi}$ (see the discussion in the comments).

In the case of $\{-1,1\}$ Do [Do24] proved that, if $R_n[-1,1]$ counts the number of roots in $[-1,1]$, then, almost surely,\[\lim_{n\to \infty}\frac{R_n[-1,1]}{\log n}=\frac{1}{\pi}.\]See also [522].

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Random polynomials with independently identically distributed coefficients are sometimes called Kac polynomials - this problem considers the case of Rademacher coefficients, i.e. independent uniform $\pm 1$ values. Erdős and Offord [EO56] showed that the number of real roots of a random degree $n$ polynomial with $\pm 1$ coefficients is $(\frac{2}{\pi}+o(1))\log n$.

There is some ambiguity whether Erdős intended the coefficients to be in $\{-1,1\}$ or $\{0,1\}$ - see the comments section.

A weaker version of this was solved by Yakir [Ya21], who proved that\[\frac{R_n}{n/2}\to 1\]in probability. (This weaker claim was also asked by Erdős, and also appears in a book of Hayman [Ha67].) More precisely,\[\lim_{n\to \infty} \mathbb{P}(\lvert R_n-n/2\rvert \geq n^{9/10}) =0.\]See also [521].

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Salem and Zygmund [SZ54] proved that $\sqrt{n\log n}$ is the right order of magnitude, but not an asymptotic.

This was settled by Halász [Ha73], who proved this is true with $C=1$.

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A problem of Salem and Zygmund [SaZy54]. Chung showed that, for almost all $t$, there exist infinitely many $n$ such that\[M_n(t) \ll \left(\frac{n}{\log\log n}\right)^{1/2}.\]Erdős (unpublished) showed that for almost all $t$ and every $\epsilon>0$ we have $\lim_{n\to \infty}M_n(t)/n^{1/2-\epsilon}=\infty$.

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Random polynomials with independently identically distributed coefficients are sometimes called Kac polynomials - this problem considers the case of Rademacher coefficients, i.e. independent uniform $\pm 1$ values. The first problem asks whether $m(f)<1$ almost surely. Littlewood [Li66] conjectured that the stronger $m(f)=o(1)$ holds almost surely.

The answer to both questions is yes: Littlewood's conjecture was solved by Kashin [Ka87], and Konyagin [Ko94] improved this to show that $m(f)\leq n^{-1/2+o(1)}$ almost surely. This is essentially best possible, since Konyagin and Schlag [KoSc99] proved that for any $\epsilon>0$\[\limsup_{n\to \infty} \mathbb
(m(f) \leq \epsilon n^{-1/2})\ll \epsilon.\]Cook and Nguyen [CoNg21] have identified the limiting distribution, proving that for any $\epsilon>0$\[\lim_{n\to \infty} \mathbb
(m(f) > \epsilon n^{-1/2}) = e^{-\epsilon \lambda}\]where $\lambda$ is an explicit constant.

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Van der Corput [Va39] proved that\[\sum_{n\leq X} \tau(f(n))\gg_f X\log X.\]Erdős [Er52b] proved using elementary methods that\[\sum_{n\leq X} \tau(f(n))\ll_f X\log X.\]Such an asymptotic formula is known whenever $f$ is an irreducible quadratic, as proved by Hooley [Ho63]. The form of $c$ depends on $f$ in a complicated fashion (see the work of McKee [Mc95], [Mc97], and [Mc99] for expressions for various types of quadratic $f$). For example,\[\sum_{n\leq x}\tau(n^2+1)=\frac{3}{\pi}x\log x+O(x).\]Tao has a blog post on this topic.

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A problem of Erdős, Herzog, and Piranian, who note that $f(z)=z^n-1$ has $\rho(f) \leq \frac{\pi/2}{n}$.

Pommerenke [Po61] proved that\[\rho(f) \geq \frac{1}{2en^2}.\]Krishnapur, Lundberg, and Ramachandran [KLR25] proved\[\rho(f) \gg \frac{1}{n\sqrt{\log n}}.\]

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A problem of Erdős, Herzog, and Piranian [EHP58], who proved that this set always has a connected component containing at least two of the roots.

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All the zeros of $f'(x)$ are all distinct real numbers in $(a_0,a_m)$ by Rolle's theorem.

This was proved by Balint [Ba60b]. Balint gives no source for the conjecture - presumably it was from Erdős in personal communication. Some generalisations of this were given by Lorch [Lo76].

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The functions $l_k(x)$ are sometimes called the fundamental functions of Lagrange interpolation, and $\Lambda$ is sometimes called the Lebesgue constant.

Faber [Fa14] proved\[\Lambda(x_1,\ldots,x_n)\gg \log n\]for all choices of $x_i$, and Bernstein [Be31] proved it is $>(\frac{2}{\pi}-o(1))\log n$. Erdős [Er61c] improved this to\[\Lambda(x_1,\ldots,x_n)> \frac{2}{\pi}\log n-O(1).\]This is best possible, since taking the $x_i$ as the roots of the $n$th Chebyshev polynomial yields\[\Lambda(x_1,\ldots,x_n)< \frac{2}{\pi}\log n+O(1).\]Erdős thought that the minimising choice is characterised by the property that the sums\[\lambda_i=\max_{x\in [x_i,x_{i+1}]}\sum_k \lvert l_k(x)\rvert\]are all equal for $0\leq i\leq n$ (where $x_0=-1$ and $x_{n+1}=1$). This conjecture was also made by Bernstein [Be31]. Kilgore and Cheney [KiCh76] proved that there exists $x_i$ for which all $\lambda_i$ are equal. Kilgore [Ki77] proved that $\Lambda$ is minimised only when all $\lambda_i$ are equal. Finally, de Boor and Pinkus [dBPi78] proved that there exists a unique minimising choice of $x_i$.

If $x_1=-1$ and $x_n=1$ then there is a unique minimising set of $x_i$, which are symmetric around $0$. (Such a choice is called canonical.) The exact minimising canonical choice is known only for $n\leq 4$. For $n=2$ the points are $-1,1$ (with $\Lambda=1$). For $n=3$ the points are $-1,0,1$ (with $\Lambda=1.25$), as shown by Bernstein [Be31]. Rack and Vajda [RaVa15] have shown that for $n=4$ the points are $-1,-t,t,1$ where $t\approx 0.4177$ is an explicit algebraic constant (with $\Lambda \approx 1.4229$).

In [Er67] Erdős suggests that an easier variant might be to have the $x_i\in \mathbb{C}$ with $\lvert x_i\rvert=1$, and seek to minimise $\max_{\lvert z\rvert=1}\sum_{k}\lvert l_k(z)\rvert$, adding it 'seems certain' that the minimising $x_i$ are the $n$th roots of unity. This was proved by Brutman [Br80] for odd $n$ and by Brutman and Pinkus [BrPi80] for even $n$.

See also [671], [1130], and [1132].

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The functions $l_k(x)$ are sometimes called the fundamental functions of Lagrange interpolation.

Erdős [Er47] could prove\[\Upsilon(x_1,\ldots,x_n)< \sqrt{n}.\]Erdős thought that the maximising choice is characterised by the property that the sums\[\lambda_i=\max_{x\in [x_i,x_{i+1}]}\sum_k \lvert l_k(x)\rvert\]are all equal for $0\leq i\leq n$ (where $x_0=-1$ and $x_{n+1}=1$), which would be the same characterisation as [1129].

This is true, and was proved by de Boor and Pinkus [dBPi78]. It follows by the bounds discussed in [1129] that\[\Upsilon(x_1,\ldots,x_n)\leq \frac{2}{\pi}\log n+O(1).\]See also [1129].

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Erdős first conjectured this minimum was achieved by taking the $x_i$ to be the roots of the integral of the Legendre polynomial, since Fejer [Fe32] had earlier shown these to be minimisers of\[\max_{x\in [-1,1]}\sum_k \lvert l_k(x)\rvert^2.\]This was disproved by Szabados [Sz66] for every $n>3$.

Erdős, Szabados, Varma, and Vértesi [ESVV94] proved that\[2-O\left(\frac{(\log n)^2}{n}\right)\leq \min I\leq 2-\frac{2}{2n-1}\]where the upper bound is witnessed by the roots of the integral of the Legendre polynomial as above.

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A result of Bernstein [Be31] implies that the set of $x\in(-1,1)$ for which\[\limsup_{n\to \infty}\frac{L_n(x)}{\log n}\geq \frac{2}{\pi}\]is everywhere dense.

Erdős [Er61c] proved that, for any fixed $x_1,\ldots,x_n\in [-1,1]$,\[\max_{x\in [-1,1]}\sum_{1\leq k\leq n}\lvert l_k(x)\rvert>\frac{2}{\pi}\log n-O(1).\]See also [1129] for more on $L_n(x)$.

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Erdős proved that, for any $C>0$, there exists $\epsilon>0$ such that if $n$ is sufficiently large and $m=\lfloor (1+\epsilon)n\rfloor$ then for any $x_1,\ldots,x_m\in [-1,1]$ there is a polynomial $P$ of degree $n$ such that $\lvert P(x_i)\rvert\leq 1$ for $1\leq i\leq m$ and\[\max_{x\in [-1,1]}\lvert P(x)\rvert>C.\]The conjectured statement would also imply this, but Erdős in [Er67] says he could not even prove it for $m=n$.

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