7 problems

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…

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…

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…

Let $n\geq 1$, and let $(k,n)$ be a valid pair for Littlewood polynomials. A pair is Littlewood-admissible if some Littlewood polynomial of degree $n$ has exactly $k$ zeros in the…

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

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 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…