10 problems

Secant-dimension conjecture. For each integer $r\geq 1$, we have

Let $\mathbb K$ be a field with characteristic $\chi(\mathbb K)\ne 2$, let $n\geq 4$, and let … be a $3$-form. Its nullity is the dimension of the largest subspace on which the ass…

Strength–slice-rank conjecture. The strength and the slice rank of a general form in $\mathcal{S}_d$ are equal.

Let $f\in \operatorname{V}_{n,d}$ be a nondegenerate form, and let $Z(f)$ denote its center. O'Ryan–Shapiro dimension conjecture. It was conjectured that … The conjecture concerns…

Let $k$ and $s$ be positive integers, and let $F$ be a form of degree $k$ with integer coefficients in $s$ variables. A non-trivial zero of $F$ over a $p$-adic field is a point in…

Let $k$ be a positive integer, and let a general form of degree $k$ in $4$ variables be a homogeneous polynomial of degree $k$. A determinant of a $k\times k$ matrix of linear form…

For a prime $p$, let $\nu_d(p)$ be the least integer $n$ such that every degree-$d$ form … has a zero in $\mathbb{Q}_p$. Artin's conjecture. For every prime $p$, … This conjecture…

Associated-form conjecture. For every ${\rm I}\in{\mathfrak I}_n^m$ there exists ${\mathbf I}\in{\mathcal I}_n^{n(m-2)}$ defined at all points of…

Let $K$ be a $\mathfrak{p}$-adic field, and let … be a system of forms in $n$ variables ${\bf x}=(x_1,\ldots,x_n)$, where $F_i$ has degree $d_i$. A zero is non-trivial if…

Isomorphism conjecture. The algebras of semi-invariants and covariants of an $n$-ary form are isomorphic.