120 problems

Fibonacci degree conjecture. Both polynomial expressions $\gamma_n(b_1^{m_1}\dots b_k^{m_k})$ and $\ell_n(b_1^{m_1}\dots b_k^{m_k})$ have degree $F_n$ in $m_j$ and degree $F_{n-1}$…

For $n\geq 0$, let $\Sigma_{k,2n}$ denote the component of weight $k+1-2n$ in the Kerov character polynomial $\Sigma_k$, and write $[R_2^i]\Sigma_{k,2n}$ for the coefficient of…

For distinct $x_1,\ldots,x_n\in[-1,1]$, let $l_k(x)=\prod_{i\ne k}\frac{x-x_i}{x_k-x_i}$ and $\lambda(x)=\sum_{k=1}^n|l_k(x)|$. Is it true that, for every fixed $-1\leq a<b\leq1$,…

Let $f(z)=\sum_{0\leq k\leq n} \epsilon_k z^k$ be a random polynomial, where $\epsilon_k\in \{-1,1\}$ independently uniformly at random for $0\leq k\leq n$. Is it true that, if…

Let $(\epsilon_k)_{k\geq 0}$ be independently uniformly chosen at random from $\{-1,1\}$. If $R_n$ counts the number of real roots of $f_n(z)=\sum_{0\leq k\leq n}\epsilon_k z^k$ th…

For every $\varepsilon>0$, for all sufficiently large integers $n$, every monic polynomial $p\in\mathbb{C}[z]$ of degree $n$ whose sublevel set … is connected satisfies, for every…

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…

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…

Let $X_n$ be the set of Chebyshev nodes and let $L(f,X_n,x)$ be the corresponding Lagrange interpolant of a continuous function $f$ at $x$. Given a fixed $x\in[-1,1]$ and a closed…

Let $C>0$. There exists $\epsilon>0$ such that if $n$ is sufficiently large the following holds. For any $x_1,\ldots,x_n\in [-1,1]$ there exist $y_1,\ldots,y_n\in [-1,1]$ such that…

Does there exist a constant $c>0$ such that, for all sufficiently large integers $n$ and every polynomial $P\in\mathbb C[z]$ satisfying … and, for every…

For $x_1,\ldots,x_n\in [-1,1]$ let $l_k(x)=\frac{\prod_{i\neq k}(x-x_i)}{\prod_{i\neq k}(x_k-x_i)},$ which are such that $l_k(x_k)=1$ and $l_k(x_i)=0$ for $i\neq k$. Let…

For $x_1,\ldots,x_n\in [-1,1]$ let $l_k(x)=\frac{\prod_{i\neq k}(x-x_i)}{\prod_{i\neq k}(x_k-x_i)},$ which are such that $l_k(x_k)=1$ and $l_k(x_i)=0$ for $i\neq k$. What is the…

For $x_1,\ldots,x_n\in [-1,1]$ let $l_k(x)=\frac{\prod_{i\neq k}(x-x_i)}{\prod_{i\neq k}(x_k-x_i)},$ which are such that $l_k(x_k)=1$ and $l_k(x_i)=0$ for $i\neq k$. Let $x_0=-1$…

For $x_1,\ldots,x_n\in [-1,1]$ let $l_k(x)=\frac{\prod_{i\neq k}(x-x_i)}{\prod_{i\neq k}(x_k-x_i)},$ which are such that $l_k(x_k)=1$ and $l_k(x_i)=0$ for $i\neq k$. Describe whi…

Among the $2^{n+1}$ polynomials $P(z)=∑_{k=0}^n ε_kz^k$ with $ε_k∈{-1,1}$, do all but $o(2^n)$ satisfy $\min_{|z|=1}|P(z)|<1$? More generally, determine the typical size of this mi…

Let $ε_0,ε_1,…$ be independent uniform random signs. Is there a constant $C>0$ such that, almost surely, $\max_{|z|=1}|∑_{k=0}^n ε_kz^k|=(C+o(1))√{n\log n}$ as $n→∞$?

For $k\geq1$, let $Q_k$ be the least possible number of nonzero coefficients of $P(x)^2$, where $P\in\mathbb{Q}[x]$ has exactly $k$ nonzero coefficients. Must $Q_k\to\infty$ as…

Is there an absolute constant $c>0$ such that every $n≥2$ and every polynomial $P(z)=∑_{k=1}^n a_kz^k$ with $|a_k|=1$ satisfy $\max_{|z|=1}|P(z)|≥(1+c)√n$?

Do there exist constants $c_1,c_2\in\mathbb{R}$ such that, for all sufficiently large integers $n$, there is a polynomial $p\in\mathbb{C}[z]$ of degree $n$ whose coefficients satis…

For $p(z)=\prod_{i=1}^n(z-z_i)$ with $|z_i|\leq1$, give a polynomial, or perhaps polylogarithmic, lower bound for the area of $\{z:|p(z)|<1\}$.

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 $p(z)=z^n-1$?

Let $\mathrm{den}(n,x)$ be the denominator polynomial associated with sums of reciprocals of subsum polynomials over partitions of $n$. Log-concavity conjecture. The sequence of co…

Let $P_n(z)$ be the sorted binomial polynomial with parameter $n$. Irreducibility conjecture. $P_n(z)$ is irreducible over $\mathbb{Q}$ for all even $n$. The analogous factorizatio…

Let $f_1,\ldots,f_k$ be irreducible integer polynomials satisfying the admissibility condition. Let $P(x)$ denote the number of integers $n\leq x$ for which all $f_i(n)$ are prime.…