12 problems

Let $n$ be a positive integer, let $\mathbb{K}$ be a field of characteristic zero, and let $f\in\mathbb{K}[x_1,\dots,x_n]$ be a homogeneous polynomial. Set…

A standard graded Artinian Gorenstein algebra over a characteristic-zero field has a symmetric Hilbert vector. Let $\mathcal{AG}_{st,\mathbb{K}}(n,d)$ denote the poset of Hilbert v…

Minimality conjecture for full Perazzo algebras. The $h$-vectors of full Perazzo algebras are minimal among the $h$-vectors of all Artinian Gorenstein algebras with the same degree…

Codimension-three Artinian Gorenstein WLP conjecture. Every such algebra $A$ has the WLP. The source notes that codimension-three standard-graded complete intersections have the WL…

The strong full Perazzo conjecture. The Hilbert vector $H$ is strongly minimal in $\mathcal{AG}(r,d)$.

Let $A$ be an Artinian Gorenstein algebra of codimension three, with socle degree $j_A$, over an infinite field $\mathsf{k}$ of characteristic $p>j_A$. For a general enough linear…

Let $M$) be an upper triangular matrix of size $d+1$ with non-negative entries. For an upper triangular matrix, let $\operatorname{diag}(i,M)$ denote its $i$-th diagonal vector, an…

Non-ubiquity conjecture. For every $r gu00e 3$, there exist algebras $A$ with decompositions $mathcal{D}(A)=(H(A)0,ldots)$ such that $mathcal{D}(A){le a}$ cannot occur as the compl…

SI-sequence conjecture. The Hilbert function $H(A)$ satisfies the additional SI condition.

Let $A=\bigoplus_i A_i$ be an Artinian Gorenstein algebra presented by quadrics over a field $\mathbb{K}$ of characteristic zero. The Weak Lefschetz Property means that there exist…

Canonical-grading conjecture. A general algebra of type $(n,d)$ is canonically graded if and only if $(n,d)$ belongs to the following list:

Associated-form conjecture. For any ${\tt I}\in{\mathcal I}_n^m$ there exists an absolute invariant $\mathbf I$ of forms of degree $n(m-2)$ on $\mathbb C^n$ such that, for all…