7 solved out of 12 shown (show only solved or open)
OPEN
If $p(z)\in\mathbb{C}[z]$ is a monic polynomial of degree $n$ then is the length of the curve $\{ z\in \mathbb{C} : \lvert p(z)\rvert=1\}$ maximised when $p(z)=z^n-1$?
A problem of Erdős, Herzog, and Piranian [EHP58].
SOLVED
If $p(z)$ is a polynomial of degree $n$ such that $\{z : \lvert p(z)\rvert\leq 1\}$ is connected then is it true that $\max_{\substack{z\in\mathbb{C}\\ \lvert p(z)\rvert\leq 1}} \lvert p'(z)\rvert \leq (\tfrac{1}{2}+o(1))n^2?$
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.

Let $p(z)=\prod_{i=1}^n (z-z_i)$ for $\lvert z_i\rvert \leq 1$. Then the area of the set where $A=\{ z: \lvert p(z)\rvert <1\}$ is $>n^{-O(1)}$ (or perhaps even $>(\log n)^{-O(1)}$).
Conjectured by Erdős, Herzog, and Piranian [ErHePi58]. 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 $\epsilon>0$.
SOLVED - $100 Let$z_i$be an infinite sequence of complex numbers such that$\lvert z_i\rvert=1$for all$i\geq 1$, and for$n\geq 1$let $p_n(z)=\prod_{i\leq n} (z-z_i).$ Let$M_n=\max_{\lvert z\rvert=1}\lvert p_n(z)\rvert$. Is it true that$\limsup M_n=\infty$? Is it true that there exists$c>0$such that for infinitely many$n$we have$M_n > n^c$, or even that for all$n$$\sum_{k\leq n}M_k > n^{1+c}?$ The weaker conjecture that$\limsup M_n=\infty$was proved by Wagner, who show that there is some$c>0$with$M_n>(\log n)^c$infinitely often. This was solved by Beck [Be91], who proved that there exists some$c>0$such that $\max_{n\leq N} M_n > N^c.$ Additional thanks to: Winston Heap SOLVED Does there exist, for all large$n$, a polynomial$P$of degree$n$, with coefficients$\pm 1$, such that $\sqrt{n} \ll \lvert P(z) \rvert \ll \sqrt{n}$ for all$\lvert z\rvert =1$, with the implied constants independent of$z$and$n$? 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]. Additional thanks to: Mehtaab Sawhney SOLVED Let$P(z)=\sum_{1\leq k\leq n}a_kz^k$for some$a_k\in \mathbb{C}$with$\lvert a_k\rvert=1$for$1\leq k\leq n$. Does there exist a constant$c>0$such that, for$n\geq 2$, we have $\max_{\lvert z\rvert=1}\lvert P(z)\rvert \geq (1+c)\sqrt{n}?$ 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]. Additional thanks to: Mehtaab Sawhney SOLVED Let$f(k)$be the minimum number of terms in$P(x)^2$, where$P\in \mathbb{Q}[x]$ranges over all polynomials with exactly$k$non-zero terms. Is it true that$f(k)\to\infty$as$k\to \infty$? 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.$ Additional thanks to: Stefan Steinerberger OPEN For any$t\in (0,1)$let$t=\sum_{k=1}^\infty \epsilon_k(t)2^{-k}$(where$\epsilon_k(t)\in \{0,1\}$). Let$R_n(t)$denote the number of real roots of$\sum_{1\leq k\leq n}\epsilon_k(t)z^k$. Is it true that, for almost all$t\in (0,1)$, we have $\lim_{n\to \infty}\frac{R_n(t)}{\log n}=\frac{\pi}{2}?$ 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]. OPEN Is it true that all except$o(2^n)$many polynomials of degree$n$with$\pm 1$-valued coefficients have$(\frac{1}{2}+o(1))n$many roots in$\{ z\in \mathbb{C} : \lvert z\rvert \leq 1\}$? For any$t\in (0,1)$let$t=\sum_{k=1}^\infty \epsilon_k(t)2^{-k}$(where$\epsilon_k(t)\in \{0,1\}$). If$S_n(t)$is the number of roots of$\sum_{1\leq k\leq n}\epsilon_k(t)z^k$in$\lvert z\rvert \leq1$then is it true that, for almost all$t\in (0,1)$, $\lim_{n\to \infty}\frac{S_n(t)}{n}=\frac{1}{2}?$ 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 [521]. Additional thanks to: Zachary Chase OPEN For any$t\in (0,1)$let$t=\sum_{k=1}^\infty \epsilon_k(t)2^{-k}$(where$\epsilon_k(t)\in \{0,1\}$). Does there exist some constant$C>0$such that, for almost all$t\in (0,1)$, $\max_{\lvert z\rvert=1}\left\lvert \sum_{k\leq n}\epsilon_k(t)z^k\right\rvert=(C+o(1))\sqrt{n\log n}?$ Salem and Zygmund [SZ54] proved that$\sqrt{n\log n}$is the right order of magnitude, but not an asymptotic. OPEN For any$t\in (0,1)$let$t=\sum_{k=1}^\infty \epsilon_k(t)2^{-k}$(where$\epsilon_k(t)\in \{0,1\}$). What is the correct order of magnitude (for almost all$t\in(0,1)$) for $M_n(t)=\max_{x\in [0,1]}\left\lvert \sum_{k\leq n}\epsilon_k(t)x^k\right\rvert?$ 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$. SOLVED Is it true that all except at most$o(2^n)$many degree$n$polynomials with$\pm 1$-valued coefficients$f(z)$have$\lvert f(z)\rvert <1$for some$\lvert z\rvert=1$? What is the behaviour of $m(f)=\min_{\lvert z\rvert=1}\lvert f(z)\rvert?$ Random polynomials with independently identically distributed coefficients are sometimes called Kac polynomials - this problem considers the case of Radamacher 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.