A problem of Erdős, Herzog, and Piranian [EHP58].

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.

Conjectured by Erdős, Herzog, and Piranian [EHP58]. The lower bound $\mu(A) \gg n^{-4}$ follows from a result of Pommerenke [Po61]. The stronger lower bound $\gg (\log n)^{-O(1)}$ is still open.

Wagner [Wa88] proves, for $n\geq 3$, the existence of such polynomials with \[\mu(A) \ll_\epsilon (\log\log n)^{-1/2+\epsilon}\] for all $\epsilon>0$.

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 \[\max_{n\leq N} M_n > N^c.\] The third question seems to remain open.

Originally a conjecture of Littlewood. The answer is yes (for all $n\geq 2$), proved by Balister, Bollobás, Morris, Sahasrabudhe, and Tiba [BBMST19].

See also [230].

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].

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 \[f(k) \gg \log k.\]

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

See also [522].

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 $(\frac{2}{\pi}+o(1))\log n$.

Solved by Yakir [Ya21], who proved that almost all such polynomials have \[\frac{n}{2}+O(n^{9/10})\] many roots in $\{ z\in \mathbb{C} : \lvert z\rvert \leq 1\}$.

See also [521].

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 $C=1$.

A problem of Salem and Zygmund [SZ54]. 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 $\lim_{n\to \infty}M_n(t)/n^{1/2-\epsilon}=\infty$.

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 $\epsilon>0$ \[\limsup_{n\to \infty} \mathbb{P}(m(f) \leq \epsilon n^{-1/2})\ll \epsilon.\] Cook and Nguyen [CoNg21] have identified the limiting distribution, proving that for any $\epsilon>0$ \[\lim_{n\to \infty} \mathbb{P}(m(f) > \epsilon n^{-1/2}) = e^{-\epsilon \lambda}\] where $\lambda$ is an explicit constant.