7 problems

Tran's conjecture. All zeros of every polynomial in the sequence lie on $\Gamma$, and these roots become dense in $\Gamma$ as $n\to\infty$.

Let $K$ be the coefficient field, and let $f(x)\in K[x]$ be a generic monic polynomial of degree $d$. Its cyclic resultants are the associated sequence of resultants. Generic cycli…

Let $K$ be the coefficient field, and let $f(x)\in K[x]$ be a reciprocal monic polynomial of even degree $d$, meaning that $f(1/x)=x^d f(x)$. Its cyclic resultants are the associat…

Sign conjecture. On the portion of this curve containing the zeros, the following inequalities hold: if $k$ is even, then

Shapiro's conjecture. Every zero of every $P_n(z)$ that is not a zero of $A(z)$ or $B(z)$ lies on $\mathcal C$. This generalizes Tran's zero-location conjecture to recurrences with…

Let $P=P(x_1,\ldots,x_{n-1})$ be a symmetric polynomial with positive coefficients that generates a period $1$ seed. Symmetric-polynomial classification conjecture. The only possib…

Let $P=P(x_1,\ldots,x_n)$ be a multilinear polynomial with positive coefficients that generates a period $1$ seed. Reversal-symmetry conjecture. … This predicts reversal symmetry f…