7 problems

Positivity conjecture. For $d\geq 2$, the trigonometric polynomials $\mathcal T_n^{(d)}(t)$ are positive for every $t\in\mathbb R$. Positivity of these polynomials is suggested as…

Let $n$ be a positive integer and let $\alpha_1,\ldots,\alpha_{2n}$ be real numbers satisfying … Consider the alternating exponential sum $\sum_{j=1}^{2n}(-1)^j e^{i\alpha_jt}$. Al…

Uniqueness conjecture. $P_n(\theta)$ is the only trigonometric polynomial of order $n$ satisfying (5.59). The claim proposes uniqueness within that class, beyond the explicitly ver…

Special-value conjecture. The following equivalences hold:

Goh–Wildberger conjecture. There are polynomials $\Phi_d(x)\in\mathbb{Z}[x]$, $d\geq 1$, such that, for every $n\geq 1$,

Let … where the coefficients $a_j$ are real and $a_1=1$. Odd conjugate-polynomial conjecture. One has … and the solution is unique, namely … This is an extremal problem for conjuga…

Bernstein-type conjecture. The function $G$ satisfies