6 problems

Poisson order-$k$ mode conjecture. Numerical evidence suggests that, for all tested $k\ge1$ and $n\ge2\kappa$,

Sturmfels's conjecture. For all integers $d,k\geq 1$, the maximum number of modes is

Carreira-Perpiñán–Williams conjecture. Among homoscedastic Gaussian mixtures in dimension $d$ with $k$ components, the maximum number of modes is

Maximum-mode conjecture. A Gaussian mixture with $k$ components in dimension $d$ has at most $k$ modes, namely

Let $f_k(x;\lambda)$ denote the probability mass function of the Poisson distribution of order $k$, and let $m_{k,\lambda}$ be the mode or modes of $f_k(x;\lambda)$, namely the val…

Let $P(x)=a_0+a_1x+\cdots+a_mx^m$ be a polynomial with nonnegative and nondecreasing coefficients. For $d>0$, let $M_*(P,d)$ and $M^*(P,d)$ denote respectively the smallest and gre…