18 problems

Let $X\subset\mathbb{R}^n$ and let $\mathcal{R}_n$ denote the polynomial ring in $n$ variables. For a set $G\subset\mathcal{R}_n$, write $G^{\mathrm{deg}(g)-1}$ for the relevant de…

For all positive integers $m,n$, the second-order Zarankiewicz number equals the recursive-line quantity: $z_2(m,n)=z_{RL}(m,n)$.

Smale's mean value conjecture.

Let $n\in\mathbb{N}$ be even, and let $\mathrm{B}^{n}=[-1,1]^n$. Write $\mathcal{Q}(\mathrm{B}^{n})_n$ for the degree-$n$ truncation of the quadratic module generated by…

For integers $p$ and $q$ with $0<q<p$, let $N(q/p)$ denote the number of elements in a minimal $(0,p)$-mediated sequence containing $q$. A sequence of integers…

Let $\theta\in[-1,1]$, let $\mathbb V^k$ be either $\mathbb Q^k$ or $\mathbb P^k$, and let $\mathbb V^k_+$ denote the corresponding cone of positive polynomials. For…

Let $\theta\in[-1,1]$ and let $\mathbb V^k$ denote either the tensor-product polynomial space $\mathbb Q^k$ or the total-degree polynomial space $\mathbb P^k$, with…

Algebraic specification conjecture. For all $n$, $[?]$ is quasi-admissible by some $x,y,z$ that are formed via algebraic specification.

Let $X\subset \mathbb{P}^n$ be a totally real variety, let $P_X$ denote its cone of nonnegative forms, and let $\mathcal{F}$ be a face of $P_X$. Face-relative-interior conjecture.…

Let a biquadratic form be a homogeneous polynomial of degree two in each of two variable vectors, and let its SOS rank be the smallest number of squares of bilinear forms in a sum-…

Let a polynomial optimization problem have a sparse moment relaxation and the corresponding dense moment relaxation, with relaxation order measured by the parameter in the respecti…

Generic finite convergence conjecture. There exists a finite set of polynomials $\varphi_1,\ldots,\varphi_L$ in the coefficients of $f$, $a$, $b$, and $c$ such that, if

Let $X \subseteq \mathbb{C}^n$ be an irreducible affine variety, and let $\overline{X} \subset \mathbb{P}^n$ be its projective closure. Let $Q_{\infty}$ denote the quadric at infin…

Let $n\ge 3$, let $m_1,\dots,m_n\in\mathbb{N}$, and let $U_1,\dots,U_n$ be independent standard Gaussian random variables. Define the polynomial $H$ on $\mathbb{R}^{n(n-1)/2}$ by ……

Let $\mathcal{G}$ and $\mathcal{H}$ be the graph objects used in the paper's polynomial model, with vertex sets $V(\mathcal{G})$ and $V(\mathcal{H})$, parameters $n_\mathcal{G}$ an…

Continuum conjecture. There is a continuum of unitarily non-equivalent MHDRs for a bivariate polynomial of degree $d$.

Let $f$ be a polynomial of degree $4$ in $\mathbb{R}[\mathrm{x}_1,\mathrm{x}_2]$ and let $g$ be a polynomial of degree $2$ in $\mathbb{R}[\mathrm{x}_1,\mathrm{x}_2]$. Suppose there…

Let $f$ be a ternary quartic, that is, a $4$-form in $\mathbb{R}[\mathrm{x}_1,\mathrm{x}_2,\mathrm{x}_3]$, and let $g$ be a quadratic form in…