7 problems

Fix integers $m\geq 1$ and $r\geq 1$. Let $T_m(N,k)(x)$ be the Hecke polynomial, with its $r$-th coefficient understood in the source's indexing. Generalized Lehmer conjecture for…

Coefficient divisibility and nonvanishing conjecture. 1. If $n=p^m$ is a power of a prime $p$, then every coefficient of $p_n$ except the coefficient of $e_1^n$ is divisible by $p$…

For a finite coefficient sequence $(a_0,\ldots,a_n)$, call it bottom heavy when $a_k\geq a_{n-k}$ for $k<n/2$. Let $S[n,k]$, $c[n,k]$, $S_B[n,k]$, and $c_B[n,k]$ be the type $A$ an…

Second Borwein conjecture. Each of $\alpha_n(q)$, $\beta_n(q)$ and $\gamma_n(q)$ has non-negative coefficients.

Let $m$ be a natural number. For a polynomial $f$, its height $h(f)$ is the maximum of the absolute values of its coefficients. Kosyak et al.'s height conjecture. For every natural…

Generalized Schur–Siegel–Smyth conjecture. Fix $\epsilon>0$. Then there exist only finitely many $f(x)\in\mathcal{F}$ such that

Let $d$ be a positive integer, let $\ol\nu\vdash d$, and let $s_{\ol\nu}(x)$ be the polynomial map associated with the staircase Kronecker-square coefficients. Let $c(\ol\nu)$ be t…