5 problems

Xu's uniqueness conjecture. For all $d$, $|T_d(x)|\le 1$, and the points in $\mathcal S_+$ and $\mathcal S_-$ are the only points on the face $x_1+\cdots+x_d=1$ at which…

For $d\ge 3$, let $T_d$ be the recursively defined polynomial in the elementary symmetric functions on the standard simplex $T^d$, and write $\|T_d\|_{C(T^d)}$ for its uniform norm…

Let $d\ge 3$, and let $p(x)$ range over polynomials of total degree less than $2d$. For the monomial $(x_1\cdots x_d)^2$, consider polynomials of the form $(x_1\cdots x_d)^2-p(x)$…

Let the Euclidean ball be the domain of approximation, and let a signature mean a signature arising in the best-approximation problem for a monomial by polynomials on that ball. Bo…

Let $u_1,\ldots,u_n$ be coordinates on $(0,\infty)^n$, and let $f\in\mathbb C[u_1,\ldots,u_n]$. Put $B=(0,\infty)^n$. Exponential integral vanishing conjecture. If … for almost all…