关于指数为$4$的代数指标以$u$-不变量为界的一个定理

On index-exponent relations over Henselian fields with local residue fields

Let $p$ be a prime number and $(K, v)$ a Henselian valued field with a residue field $\widehat K$. This paper determines the Brauer $p$-dimension of $K$, in case $p \neq {\rm char}(\widehat K)$ and $\widehat K$ is a $p$-quasilocal field properly included in its maximal $p$-extension. When $\widehat K$ is a local field, it describes index-exponent pairs of central division $K$-algebras of $p$-primary degrees. The same goal is achieved, if $(K, v)$ is maximally complete, char$(K) = p$ and $\widehat K$ is local.

Period-index and u-invariant questions for function fields over complete discretely valued fields

Let K be a complete discretely valued field and F the function field of a curve over K. If the characteristic of the residue field k of K is p > 0, then we give a bound for the Brauer p-simension of F in terms of the p-rank of k. If k is a perfect field of characteristic 2, we show that the u-invaraint of F is at most 8.

On Brauer $p$-dimensions and index-exponent relations over finitely-generated field extensions

Let $E$ be a field of absolute Brauer dimension abrd$(E)$, and $F/E$ a transcendental finitely-generated extension. This paper shows that the Brauer dimension Brd$(F)$ is infinite, if abrd$(E) = \infty $. When the absolute Brauer $p$-dimension abrd$_{p}(E)$ is infinite, for some prime number $p$, it proves that for each pair $(n, m)$ of integers with $n \ge m > 0$, there is a central division $F$-algebra of Schur index $p ^{n}$ and exponent $p ^{m}$. Lower bounds on the Brauer $p$-dimension Brd$_{p}(F)$ are obtained in some important special cases where abrd$_{p}(E) < \infty $. These results s…

通用图序列、色函数及其指标理论

Universal graph series, chromatic functions, and their index theory

该研究引入通用图序列概念，提出四个图不变量并探讨性质，其中一个不变量推广了色对称函数且为图的完全不变量，为图论不变量研究提供新方向。

幂零李代数的指数

The index of nilpotent Lie algebras

本文研究多类幂零李代数的指数，并针对自由幂零、图及丝状幂零李代数给出显式计算公式。