12 problems

Let $d$, $k$, and $s$ be positive integers, and let $p_1,\ldots,p_k\in\mathbb{R}[x_1,\ldots,x_d]$ be homogeneous polynomials of the same degree $s$, no two of which are linearly de…

Nullstellensatz formulation of the Casas–Alvero conjecture. The conjecture is equivalent to

Casas–Alvero conjecture. If $\operatorname{char} K=0$ and $f\in K[x]$ is a Casas–Alvero polynomial of degree $d$ with $a_d=0$, then

Degree conjecture. For every relevant $n$,

Relatively Prime Conjecture. For all such ${\bf a},{\bf b}$ and all $n\in\mathbb N$, the polynomials $P_{{\bf a},{\bf b},n}(X)$ and $P_{{\bf a},{\bf b},n+1}(X)$ have no common zero…

Strong Factorial Conjecture. $F_\cap^{[m]}={\mathbb C}^{[m]}$; equivalently, every polynomial belongs to the strong factorial set. This conjecture is explicitly stated as stronger…

Factorial Conjecture. For all $f\in{\mathbb C}^{[m]}$,

Let $\xi_1,\dotsc,\xi_n\subset\mathbb{R}^{3,1}$ be $n$ non-intersecting world lines in Minkowski space, and let $x_i\in\xi_i$ be distinct events. For $i\ne j$, let $u_{ij}$ be the…

Hyperbolic conjecture. For every such configuration, the polynomials $\beta_1(z),\dotsc,\beta_n(z)$ are linearly independent over $\mathbb{C}$. This is presented as a stronger vers…

Euclidean conjecture. For every such configuration, the polynomials $\beta_1(z),\dotsc,\beta_n(z)$ are linearly independent over $\mathbb{C}$. This is the first and simplest versio…

Let $K$ be a field of characteristic zero, let ${\mathcal A}_K[\xi,z]$ denote the polynomial algebra in the variables $\xi$ and $z$, and let $P(\xi,z)\in {\mathcal A}_K[\xi,z]$. Le…

Let $K$ be a field of characteristic zero, let $P(z)\in {\mathcal A}_K[z]$, and let $\Lambda$ be a quadratic differential operator on ${\mathcal A}_K[z]$. Assume … for every intege…