Finitistic Extension Degree

不存在具有素数度域扩张的Keller映射

There are no Keller maps having prime degree field extensions

本文证明了不存在使域扩张次数为素数的Keller映射，从而支持二维Jacobian猜想。

On the integral degree of integral ring extensions

Let $A\subset B$ be an integral ring extension of integral domains with fields of fractions $K$ and $L$, respectively. The integral degree of $A\subset B$, denoted by ${\rm d}_A(B)$, is defined as the supremum of the degrees of minimal integral equations of elements of $B$ over $A$. It is an invariant that lies in between ${\rm d}_K(L)$ and $μ_A(B)$, the minimal number of generators of the $A$-module $B$. Our purpose is to study this invariant. We prove that it is sub-multiplicative and upper-semicontinuous in the following three cases: if $A\subset B$ is simple; if $A\subset B$ is projective …

The relative approximation degree in valued function fields

We continue the work of Kaplansky on immediate valued field extensions and determine special properties of elements in such extensions. In particular, we are interested in the question when an immediate valued function field of transcendence degree 1 is henselian rational (i.e., generated, modulo henselization, by one element). If so, then wild ramification can be eliminated in this valued function field. The results presented in this paper are crucial for the first author's proof of henselian rationality over tame fields, which in turn is used in his work on local uniformization.

任意赋值环的 p 次扩张与“最佳 f”

Degree $p$ Extensions of Arbitrary Valuation Rings and "Best $f$"

本文给出了 Henselian 赋值环上 p 次 Artin-Schreier 与 Kummer 扩张中“最佳 f”的显式刻画，并将其推广到更高秩赋值环的缺陷扩张情形，完善了 p 次扩张的基础理论。

Valued fields with finitely many defect extensions of prime degree

We prove that a valued field of positive characteristic $p$ that has only finitely many distinct Artin-Schreier extensions (which is a property of infinite NTP$_2$ fields) is dense in its perfect hull. As a consequence, it is a deeply ramified field and has $p$-divisible value group and perfect residue field. Further, we prove a partial analogue for valued fields of mixed characteristic and observe an open problem about 1-units in this setting. Finally, we fill a gap that occurred in a proof in an earlier paper in which we first introduced a classification of Artin-Schreier defect extensions.

关于数域上整矩阵相似的扩张次数界

On Bounds of Extension Degrees for Similarity of Integral Matrices over Number Fields

证明了数域K上n×n整矩阵在K的整数环所有完备化上相似时，不存在统一的扩张次数界，但给出了给定可分特征多项式时的上界。

On the numerical range of square matrices with coefficients in a degree 2 Galois field extension

Let $L$ be a degree $2$ Galois extension of the field $K$ and $M$ an $n\times n$ matrix with coefficients in $L$. Let $\langle \ ,\ \rangle : L^n\times L^n\to L$ be the sesquilinear form associated to the involution $L\to L$ fixing $K$. We use $\langle \ ,\ \rangle$ to define the numerical range $\mathrm{Num} (M)$ of $M$ (a subset of $L$), extending the classical case $K=\mathbb {R}$, $L=\mathbb {C}$ and the case of a finite field introduced by Coons, Jenkins, Knowles, Luke and Rault. There are big differences with respect to both cases for number fields and for all fields in which the image o…

The determination of integral closures and geometric applications

We express explicitly the integral closures of some ring extensions; this is done for all Bring-Jerrard extensions of any degree as well as for all general extensions of degree < 6; so far such an explicit expression is known only for degree < 4 extensions. As a geometric application we present explicitly the structure sheaf of every Bring-Jerrard covering space in terms of coefficients of the equation defining the covering; in particular, we show that a degree-3 morphism f : Y --> X is quasi-etale if and only if the first Chern class of the sheaf f_*(O_Y) is trivial (details in Theorem 5.3). …