16 solved out of 33 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 Let $f(\theta) = \sum_{k\geq 1}c_k e^{ik\theta}$ be a trigonometric polynomial (so that the$c_k\in \mathbb{C}$are finitely supported) with real roots such that$\max_{\theta\in [0,2\pi]}\lvert f(\theta)\rvert=1$. Then $\int_0^{2\pi}\lvert f(\theta)\rvert \mathrm{d}\theta \leq 4.$ This was solved independently by Kristiansen [Kr74] (only in the case when$c_k\in\mathbb{R}$) and Saff and Sheil-Small [SSS73] (for general$c_k\in \mathbb{C}$). Additional thanks to: Winston Heap, Vjekoslav Kovac, Karlo Lelas SOLVED Is there an entire non-linear function$f$such that, for all$x\in\mathbb{R}$,$x$is rational if and only if$f(x)$is? More generally, if$A,B\subseteq \mathbb{R}$are two countable dense sets then is there an entire function such that$f(A)=B$? Solved by Barth and Schneider [BaSc70], who proved that if$A,B\subset\mathbb{R}$are countable dense sets then there exists a transcendental entire function$f$such that$f(z)\in B$if and only if$z\in A$. In [BaSc71] they proved the same result for countable dense subsets of$\mathbb{C}$. Additional thanks to: Boris Alexeev, Dustin Mixon, and Terence Tao SOLVED Let$f=\sum_{n=0}^\infty a_nz^n$be an entire function. Is it true that if $\lim_{r\to \infty} \frac{\max_n\lvert a_nr^n\rvert}{\max_{\lvert z\rvert=r}\lvert f(z)\rvert}$ exists then it must be$0$? Clunie (unpublished) proved this if$a_n\geq 0$for all$n$. This was disproved in general by Clunie and Hayman [ClHa64], who showed that the limit can take any value in$[0,1/2]$. See also [513]. 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$(S_n)_{n\geq 1}$be a sequence of sets of complex numbers, none of which have a finite limit point. Does there exist an entire function$f(z)$such that, for all$n\geq 1$, there exists some$k_n\geq 0$such that $f^{(k_n)}(z) = 0\textrm{ for all }z\in S_n?$ Solved in the affirmative by Barth and Schneider [BaSc72]. Additional thanks to: Zachary Chase and Terence Tao 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 OPEN Let$n\geq 1$and$f(n)$be maximal such that, for every$a_1\leq \cdots \leq a_n\in \mathbb{N}$we have $\max_{\lvert z\rvert=1}\left\lvert \prod_{i}(1-z^{a_i})\right\rvert\geq f(n).$ Estimate$f(n)$- in particular, is it true that there exists some constant$c>0$such that $f(n) \geq \exp(n^{c})?$ Erdős and Szekeres [ErSz59] proved that$\lim f(n)^{1/n}=1$and$f(n)>\sqrt{2n}$. Erdős proved an upper bound of$f(n) < \exp(n^{1-c})$for some constant$c>0$with probabilistic methods. Atkinson [At61] showed that$f(n) <\exp(cn^{1/2}\log n)$for some constant$c>0$. This was improved to $f(n) \leq \exp( cn^{1/3}(\log n)^{4/3})$ by Odlyzko [Od82]. If we denote by$f^*(n)$the analogous quantity with the assumption that$a_1<\cdots<a_n$then Bourgain and Chang [BoCh18] prove that $f^*(n)< \exp(c(n\log n)^{1/2}\log\log n).$ Additional thanks to: Zachary Chase, Stefan Steinerberger SOLVED If$z_1,\ldots,z_n\in \mathbb{C}$with$\lvert z_i\rvert=1$then is it true that the probability that $\lvert \epsilon_1z_1+\cdots+\epsilon_nz_n\rvert \leq \sqrt{2},$ where$\epsilon_i\in \{-1,1\}$uniformly at random, is$\gg 1/n$? A reverse Littlewood-Offord problem. Erdős originally asked this with$\sqrt{2}$replaced by$1$, but Carnielli and Carolino [CaCa11] observed that this is false, choosing$z_1=1$and$z_k=i$for$2\leq k\leq n$, where$n$is even, since then the sum is at least$\sqrt{2}$always. Solved in the affirmative by He, Juškevičius, Narayanan, and Spiro [HJNS24]. The bound of$1/n$is the best possible, as shown by taking$z_k=1$for$1\leq k\leq n/2$and$z_k=i$otherwise. See also [498]. Additional thanks to: Zach Hunter 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\}$). 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}?$ See also [521] and [522]. 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 Let$A\subset \mathbb{C}$be a finite set of fixed size, for any$k\geq 1$let $A_k = \{ z_1+\cdots+z_k : z_i\in A\textrm{ distinct}\}.$ For$k>2$does the set$A_k$(together with the size of$A$) uniquely determine the set$A$? A problem of Selfridge and Straus [SeSt58], who prove that this is true if$k=2$and$\lvert A\rvert \neq 2^l$(for$l\geq 0$). On the other hand, there are examples with two distinct$A,B$both of size$2^l$such that$A_2=B_2$. More generally, they prove that$A$is uniquely determined by$A_k$if$n$is divisible by a prime greater than$k$. Selfridge and Straus sound more cautious than Erdős, and it may well be that for all$k>2$there exist$A,B$of the same size with identical$A_k=B_k$. (In [Er61] Erdős states this problem incorrectly, replacing sums with products. This product formulation is easily seen to be false, as observed by Steinerberger: consider the case$k=3$and subsets of the 6th roots of unity corresponding to$\{0,1,2,4\}$and$\{0,2,3,4\}$(as subsets of$\mathbb{Z}/6\mathbb{Z}$). The correct problem statement can be found in the paper of Selfridge and Straus that Erdős cites.) Additional thanks to: Stefan Steinerberger SOLVED Let$z_1,\ldots,z_n\in\mathbb{C}$with$1\leq \lvert z_i\rvert$for$1\leq i\leq n$. Let$D$be an arbitrary disc of radius$1$. Is it true that the number of sums of the shape $\sum_{i=1}^n\epsilon_iz_i \textrm{ for }\epsilon_i\in \{-1,1\}$ which lie in$D$is at most$\binom{n}{\lfloor n/2\rfloor}$? A strong form of the Littlewood-Offord problem. Erdős [Er45] proved this is true if$z_i\in\mathbb{R}$, and for general$z_i\in\mathbb{C}$proved a weaker upper bound of $\ll \frac{2^n}{\sqrt{n}}.$ This was solved in the affirmative by Kleitman [Kl65], who also later generalised this to arbitrary Hilbert spaces [Kl70]. See also [395]. Additional thanks to: Stijn Cambie OPEN Let$f(z)\in\mathbb{C}[z]$be a monic non-constant polynomial. Can the set $\{ z\in \mathbb{C} : \lvert f(z)\rvert \leq 1\}$ be covered by a set of circles the sum of whose radii is$\leq 2$? 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 (in fact the entire set is covered by a single circle with radius$2$). OPEN If$A\subset \mathbb{Z}$is a finite set of size$N$then is there some absolute constant$c>0$and$\theta$such that $\sum_{n\in A}\cos(n\theta) < -cN^{1/2}?$ Chowla's cosine problem. The best known bound currently, due to Ruzsa [Ru04] (improving on an earlier result of Bourgain [Bo86]), replaces$N^{1/2}$by $\exp(O(\sqrt{\log N}).$ The example$A=B-B$, where$B$is a Sidon set, shows that$N^{1/2}$would be the best possible here. OPEN Let$f(z)\in \mathbb{C}[z]$be a monic polynomial of degree$n$and $A = \{ z\in \mathbb{C} : \lvert f(z)\rvert\leq 1\}.$ Is it true that, for every such$f$and constant$c>0$, the set$A$can have at most$O_c(1)$many components of diameter$>1+c$(where the implied constant is in particular independent of$n$)? SOLVED Is it true that, if$A\subset \mathbb{Z}$is a finite set of size$N$, then $\int_0^1 \left\lvert \sum_{n\in A}e(n\theta)\right\rvert \mathrm{d}\theta \gg \log N,$ where$e(x)=e^{2\pi ix }$? Littlewood's conjecture, proved independently by Konyagin [Ko81] and McGehee, Pigno, and Smith [MPS81]. OPEN Let$f=\sum_{n=0}^\infty a_nz^n$be an entire function. What is the greatest possible value of $\liminf_{r\to \infty} \frac{\max_n\lvert a_nr^n\rvert}{\max_{\lvert z\rvert=r}\lvert f(z)\rvert}?$ It is trivial that this value is in$[1/2,1)$. Kövári (unpublished) observed that it must be$>1/2$. Clunie and Hayman [ClHa64] showed that it is$\leq 2/\pi-c$for some absolute constant$c>0$. Some other results on this quantity were established by Gray and Shah [GrSh63]. See also [227]. OPEN Let$f(z)$be an entire function. Does there exist a path$L$so that, for every$n$, $\lvert f(z)/z^n\rvert \to \infty$ as$z\to \infty$along$L$? Can the length of this path be estimated in terms of$M(r)=\max_{\lvert z\rvert=r}\lvert f(z)\rvert$? Does there exist a path along which$\lvert f(z)\rvert$tends to$\infty$faster than a fixed function of$M(r)$(such that$M(r)^\epsilon$)? Boas (unpublished) has proved the first part, that such a path must exist. OPEN Let$f(z)$be an entire function, not a polynomial. Does there exist a path$C$such that, for every$\lambda>0$, the integral $\int_C \lvert f(z)\rvert^{-\lambda} \mathrm{d}z$ is finite? Huber [Hu57] proved that for every$\lambda>0$there is a path$C_\lambda$such that this integral is finite. OPEN Let$f(z)=\sum_{k\geq 1}a_k z^{n_k}$be an entire function of finite order such that$\lim n_k/k=\infty$. Let$M(r)=\max_{\lvert z\rvert=r}\lvert f(z)\rvert$and$m(r)=\max_n \lvert a_nr^n\rvert$. Is it true that $\limsup\frac{\log m(r)}{\log M(r)}=1?$ A problem of Pólya. Results of Wiman [Wi14] imply that if$(n_{k+1}-n_k)^2>n_k$then$\limsup \frac{m(r)}{M(r)}=1$. Erdős and Macintyre [ErMa54] proved this under the assumption that $\sum_{k\geq 2}\frac{1}{n_{k+1}-n_k}<\infty.$ OPEN Let$f(z)=\sum_{k=1}^\infty a_kz^{n_k}$be an entire function. Is it true that if$n_k/k\to \infty$then$f(z)$assumes every value infinitely often? A conjecture of Fejér and Pólya. Fejér [Fe08] proved that if$\sum\frac{1}{n_k}<\infty$then$f(z)$assumes every value at least once, and Biernacki [Bi28] showed that this holds under the assumption that$n_k/k\to \infty$. SOLVED Let$z_1,\ldots,z_n\in \mathbb{C}$with$z_1=1$. Must there exist an absolute constant$c>0$such that $\max_{1\leq k\leq n}\left\lvert \sum_{i}z_i^k\right\rvert>c?$ A problem of Turán, who proved that this maximum is$\gg 1/n$. This was solved by Atkinson [At61b], who showed that$c=1/6$suffices. This has been improved by Biró, first to$c=1/2$[Bi94], and later to an absolute constant$c>1/2$[Bi00]. Based on computational evidence it is likely that the optimal value of$c$is$\approx 0.7$. 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]. SOLVED 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\}$? 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 [474] and [521]. Additional thanks to: Michal Bassan and 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. Additional thanks to: Mehtaab Sawhney OPEN Let$a_n\in \mathbb{R}$be such that$\sum_n \lvert a_n\rvert^2=\infty$and$\lvert a_n\rvert=o(1/\sqrt{n})$. Is it true that, for almost all$\epsilon_n=\pm 1$, there exists some$z$with$\lvert z\rvert=1$(depending on the choice of signs) such that $\sum_n \epsilon_n a_n z^n$ converges? It is unclear to me whether Erdős also intended to assume that$\lvert a_{n+1}\rvert\leq \lvert a_n\rvert$. It is 'well known' that, for almost all$\epsilon_n=\pm 1$, the series diverges for almost all$\lvert z\rvert=1$(assuming only$\sum \lvert a_n\rvert^2=\infty$). Dvoretzky and Erdős [DE59] showed that if$\lvert a_n\rvert >c/\sqrt{n}$then, for almost all$\epsilon_n=\pm 1$, the series diverges for all$\lvert z\rvert=1$. OPEN Given$a_{i}^n\in [-1,1]$for all$1\leq i\leq n<\infty$we define$p_{i}^n$as the unique polynomial of degree$n-1$such that$p_{i}^n(a_{i}^n)=1$and$p_{i}^n(a_{i'}^n)=0$if$1\leq i'\leq n$with$i\neq i'$. We similarly define $\mathcal{L}^nf(x) = \sum_{1\leq i\leq n}f(a_i^n)p_i^n(x),$ the unique polynomial of degree$n-1$which agrees with$f$on$a_i^n$for$1\leq i\leq n$. Is there such a sequence of$a_i^n$such that for every continuous$f:[-1,1]\to \mathbb{R}$there exists some$x\in [-1,1]$where $\limsup_{n\to \infty} \sum_{1\leq i\leq n}\lvert p_{i}^n(x)\rvert=\infty$ and yet $\mathcal{L}^nf(x) \to f(x)?$ Bernstein [Be31] proved that for any choice of$a_i^n$there exists$x_0\in [-1,1]$such that $\limsup_{n\to \infty} \sum_{1\leq i\leq n}\lvert p_{i}^n(x)\rvert=\infty.$ Erdős and Vértesi [ErVe80] proved that for any choice of$a_i^n$there exists a continuous$f:[-1,1]\to \mathbb{R}$such that $\limsup_{n\to \infty} \lvert \mathcal{L}^nf(x)\rvert=\infty$ for almost all$x\in [-1,1]\$.