13 problems

Let $f\in\mathbb{C}[x_0,\dots,x_n]$ and $g\in\mathbb{C}[y_0,\dots,y_m]$ be polynomials of the same degree $d$, and let $\operatorname{rk}$ denote their Waring rank. Symmetric Stras…

Fix $d$, and let $k\in\mathbb{N}$ be a constant depending only on $d$. In the setting of Costello's conjecture, let $f\in\mathbb{F}[x_1,\dots,x_n]$ and let $\xi_1,\dots,\xi_n$ be i…

Let ${\mathbf{k}}$ be a non-closed field, let $\overline{{\mathbf{k}}}$ be an algebraic closure of ${\mathbf{k}}$, and let $f\in {\mathbf{k}}[x_1,\ldots,x_n]$ be a homogeneous poly…

Rank-separation conjecture. There exist no functions $f,g,h\colon\mathbb{N}\to\mathbb{N}$ such that, for every $p\in\mathbb{R}[x,y]$, independently of the degree of $p$,

Monic Shapiro conjecture. The addition map satisfies

Shapiro's conjecture. Every binary form of degree $d e$ can be written as the sum of $d$ $d$-th powers of forms of degree $e$.

Strassen's decomposition conjecture. If $F=\sum_{i=1}^sF_i\in S$ is a degree $d\geq3$ form such that $F_i\in S^{[i]}$ for all $i=1,\ldots,s$, then any minimal Waring decomposition…

Thom's conjecture. Every germ at $0$ of an analytic function $F$ has a unique finest decomposition as a sum of germs of functions in independent variables, up to analytic equivalen…

Let $p$ be a general binary form of even degree $2s$, and let $\alpha_j,\beta_j\in\mathbb{C}$ for $1\leq j\leq s+1$. Canonical representation conjecture. The form $p$ can be writte…

Classification conjecture. Every maximal $i$-collision with $i\geq 2$ at degree $p^{2}$ is either a Frobenius collision or has the form described in Corollary autoref{cor:transform…

Completeness conjecture. These projectively constructed collisions are all the possibilities for general polynomial collisions.

Completeness conjecture. The presented decompositions comprise all decompositions of the prescribed shape for non-additive polynomials.

Let $\mathbb{K}$ be a field of characteristic $\operatorname{char}(\mathbb{K})$, and let $f$ be a rational function over $\mathbb{K}$ whose degree is not a multiple of…