13 problems

Pfister-neighbour conjecture. If $s<\frac{2^{n+1}}{3}$ and $\varphi_1$ is defined over $F$, then $\varphi$ is a Pfister neighbour.

Let $L/K$ be a finite Galois field extension, and suppose that $K$ contains a primitive fourth root of unity. The trace form is the non-degenerate quadratic form … defined over $K$…

Vishik–Hoffmann conjectures. The following assertions should hold:

Hoffmann–Laghribi's conjecture. If $s<2r$ and $\varphi_1$ is defined over $F$, then $\varphi$ is a Pfister neighbour.

Let $F$ be a field, let $I^n(F)$ denote the $n$th power of the fundamental ideal in the Witt ring of $F$, and let $u$-fold Pfister forms be tensor products of $u$ binary forms. A…

Let $R$ be a local regular ring with fraction field $K$, let $Q$ be a non-degenerate quadratic form over $R$, and let $c\in R$ be invertible. The equation $Q=c$ means that the quad…

Let $F$ be a field of characteristic different from $2$, and let $I^nF$ denote the $n$-th power of the fundamental ideal in the Witt ring of $F$. Say that $I^nF$ is $m$-linked if e…

Pfister involution conjecture.

Izhboldin–Vishik conjecture. The dimensions of such forms necessarily belong to these intervals.

Characteristic-two height-two conjecture. There exist a $(d-1)$-fold Pfister form $\pi$ and a $4$-dimensional form $\mathfrak{c}$ of nontrivial determinant such that

Expected height-two classification. Exactly one of the following holds:

Hoffmann's maximal splitting conjecture. If $q$ has maximal splitting, then $q$ is a Pfister neighbour.

Let $F$ be a field of characteristic different from $2$, and let $n\geq 2$. Say that $F$ has property $PN(n)$ if every quadratic form over $F$ of dimension $2^n+1$ is a Pfister nei…