8 problems

Hawaiian conjecture. Each chord of $\mathcal{G}_{P}$ contains at least one nonreal zero of $P$.

Craven–Csordas–Smith conjecture. The polynomial $W(P,P')$ has at most $2s$ real zeros.

Let $p(x)$ be a real polynomial of even degree $n$. For a real polynomial $q$, write $\sharp_r q$ for the number of its real zeros. Shapiro's Conjecture 12. … The conjecture concer…

Strengthened Hawaii conjecture. There exists $\sigma\in\mathbb{R}$ such that

Let $P(x)=\sum_{k=0}^n a_kx^k$ be a polynomial with positive coefficients. Define … where $a_{-1}=a_{n+1}=0$. Let $0=k_1<k_2<\cdots<k_m=n$ be the sequence of all indices $k_i$ for…

Let $P(x)=\sum_{k=0}^n a_kx^k$ be a polynomial with positive coefficients. Define its weighted tropical polynomial by … A corner is a point where this piecewise-linear function is…

Generalized Hawaii conjecture. If $\varkappa>\dfrac{n-1}{n}$, then

Let $f\colon \mathbb{R}^n\to\mathbb{R}$ be a real polynomial. Denote by $\mathcal{B}_{\infty}(f)$ the bifurcation set at infinity and by $\mathcal{S}_{\infty}(f)$ the set of asympt…