21 problems

Golay's skew-symmetric reduction conjecture.

Günther–Schmidt conjecture. For every $k e 5$, the function $phik(alpha)$ attains its minimum at $alpha=frac14$.

Unbounded unimodular zeros conjecture.

Let $P$ be a Littlewood polynomial of the form … where $\epsilon_j\in\{-1,1\}$ and $z\in S^1$. Newman's conjecture. There exists a constant $c'>0$ such that … The source attributes…

Let $\beta$ be a parameter in the construction of generalized Turyn polynomials, and let the normalized Mahler measure refer to the Mahler measure divided by the square-root normal…

Let $n>4$, and let … be a polynomial with coefficients $a_i\in\{-1,1\}$. Ultra-flatness conjecture at roots of unity. There exists $\varepsilon_0>0$ such that … This is a sampling…

Let $P(z)=\sum_{n=0}^N a_nz^n$ be a Littlewood polynomial of degree $N$, with $a_n\in\{\pm1\}$, and call it reciprocal if $P(z)=z^NP(1/z)$. Let $Z_{\mathcal{L}}(N)$ be the minimum…

Let $f(X)=\sum_{i=0}^n \varepsilon_iX^i$ be sampled uniformly at random from the degree-$n$ Littlewood polynomials, where the coefficients $\varepsilon_i$ are independent and unifo…

Symmetry conjecture. If there is an LP of length $n$ and exact order $m$, then there is one that is $(-1)^m$-symmetric. Alternatively, if there is an equipowerful bipartion of leng…

Littlewood's conjecture. For every $\epsilon>0$, there are infinitely many $n$ for which there exists $f\in\mathcal{L}_n$ such that

Erdős's conjecture. There is a positive constant $c_1$ such that

Let $\mathcal L_n$ be the family of Littlewood polynomials of degree $n$. A polynomial $f(z)\in\mathcal L_n$ is skew-symmetric if its coefficients satisfy the skew-symmetry conditi…

E-transformation conjecture. For any $P_1(x)\in LC(N,i)$, there exists $P_2(x)\in LC(N,i)$ such that

Littlewood's conjecture. There exist positive constants $c_1$ and $c_2$ such that, for every $n$, one can choose $p_n$ satisfying

Borwein and Choi's conjecture. The polynomial $P(x)$ is cyclotomic if and only if

A Littlewood polynomial is an analytic polynomial with coefficients in $\{+1,-1\}$; assume $P$ is $L^2$-normalized. Erdős's conjecture. For every such polynomial $P$, … for some co…

Let $\mathcal{L}=\bigcup_{N=1}^{+\infty}\{+1,-1\}^N$ be the class of Littlewood sequences, and let $U_N(z)$ denote the associated $L^2$-normalized analytic polynomial for a sequenc…

A Littlewood polynomial is a polynomial $f(z)$ in $\mathbb{C}[z]$ whose coefficients all belong to $\{-1,1\}$. For a polynomial on the complex unit circle, write … A polynomial is…

A Littlewood polynomial is an analytic polynomial whose coefficients are all $7$. For a polynomial $P$ of degree $n$, write … for its $L^1$ norm on the unit circle. Newman's conjec…

Let $f$ range over Littlewood polynomials, meaning polynomials whose coefficients belong to $\{-1,1\}$. Consider the ratio $||f||_4/||f||_2$ as $\deg(f)\to\infty$. Golay's conjectu…

Let $f$ range over Littlewood polynomials, meaning polynomials whose coefficients belong to $\{-1,1\}$. Consider the ratio $||f||_4/||f||_2$ as $\deg(f)\to\infty$. Høholdt–Jensen c…