16 problems

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…

Hawaii conjecture. The number of real critical points of the logarithmic derivative $Q[p]$ satisfies the lower bound stated by Craven, Csordas, and Smith.

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.

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

Boris Shapiro's second conjecture.

Let $p$ be a real polynomial of degree $n\geq 2$, and let $Z_\mathbb{C}(p)$ denote the number of its non-real zeros, counted with multiplicity. For a real rational function, let…

Let $P_n$ be a polynomial of order $n$ with $P_{n-1}=P'_n/n$, and suppose … Let $alpha^{(n-1)}{i}$, for $i=1,ldots,n-1$, be the distinct real roots of $P{n-1}$, let $Delta(Pn)$ d…

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…

Special-case generalized Hawaii conjecture. The following estimates hold:

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

Hawaii conjecture. The number of real zeros of $Q_1[p]$ does not exceed the number of non-real zeros of $p$.

Let $P$ be a real polynomial of degree $d$ with all coefficients nonzero. Suppose its coefficient sequence has $c$ sign changes and $p$ sign preservations, so that $c+p=d$. Write…

Realizability conjecture. The only pairs $(pos,neg)$ that can be non-realizable have either $pos=0$ or $neg=0$. Equivalently, every pair $(pos,neg)$ satisfying the standard conditi…

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…