13 problems

Let $L\subseteq\mathbb{P}^n$ be a linear subspace of dimension $d$ not contained in any coordinate hyperplane, let $M=\operatorname{M}(L)$, and let $F$ be a flat of $M$. Write…

Let $n=qp$ and let $I=\langle f_0,f_1,\ldots,f_{p-1}\rangle\cap R^{S_n}$ be the elimination ideal, where the $f_i$ are the polynomials obtained by collecting the coefficients of th…

Rojas's gcd conjecture. One should be able to pick out the isolated roots simply by computing the gcd of the resultants arising from two different generic perturbations.

Dilation elimination conjecture. One has

Elimination ideal conjecture. One has

Let $f(x_1,\ldots,x_n)$ be a generic form, and let $\Delta(f)$ denote its multivariate discriminant. Let ${\tt Hp}(f,[x_n,\ldots,x_1])$ be the polynomial associated with the iterat…

Recursive elimination conjecture. For arbitrary $n$,

Let $f_1,f_2$ define two affine plane curves, and let $g$ generate their elimination ideal in the $y$-coordinate, while $R$ denotes their resultant with respect to $x$. Assume that…

Let $\mathcal{I}_{{\mathbf{u}}}$ be the elimination ideal associated with a Laurent transformally essential system, and let ${\mathbf{R}}$ be the sparse difference resultant of…

Let $n$ be the ambient dimension, and let ${\cal F}(z)$ denote the polynomial constructed in the problems of Sect., with the two constructions given in equations (11) and (12). In…

Let $R=K[x_0,\ldots,x_n]$, let ${\mathfrak p}\in\operatorname{mProj}(R)$, let ${\mathfrak a}\subseteq R$ be a graded ideal with ${\mathfrak a}\not\subseteq{\mathfrak p}$, and let…

Silvestrov's conjecture. Two commuting elements in the general algebra $H_K(q)$ should lie on an algebraic curve, and the eliminant construction of Burchnall and Chaundy should pro…

Arbitrary-degree elimination conjecture. For arbitrary $n$, $L$ has projective dimension two and specializes to the defining ideal of the Rees algebra $\mathcal R$.