论文标题
当地动机Brouwer学位的代表性
Representability of the local motivic Brouwer degree
论文作者
论文摘要
我们研究哪种二次形式可表示为地图$ f的本地程度：我们的主要观察结果是，在某些基本场上$ K $，并非所有二次形式都可以作为当地学位表示。从经验上讲，地图$ f的本地程度：a^n \ to a^n $具有许多双曲线求和，我们证明实际上低级的本地程度就是这种情况。我们建立了最多7美元的二次排名形式的完整分类，这些分类在所有特征的基本场上都可以作为当地程度，不同于$ 2 $。 Eisenbud和Levine还研究了双曲线汇总的数量，在那里他们建立了必须以二次形式出现的双曲线形式的界限，这些形式必须以局部程度表示。对于等级5的本地程度，我们的证明方法是基本和建设性的，而Eisenbud和Levine的工作更为笼统。我们提供了进一步的例子家庭，这些示例验证了艾森布德和莱文在某些情况下的界限很紧。
We study which quadratic forms are representable as the local degree of a map $f : A^n \to A^n$ with an isolated zero at $0$, following the work of Kass and Wickelgren who established the connection to the quadratic form of Eisenbud, Khimshiashvili, and Levine. Our main observation is that over some base fields $k$, not all quadratic forms are representable as a local degree. Empirically the local degree of a map $f : A^n \to A^n$ has many hyperbolic summands, and we prove that in fact this is the case for local degrees of low rank. We establish a complete classification of the quadratic forms of rank at most $7$ that are representable as the local degree of a map over all base fields of characteristic different from $2$. The number of hyperbolic summands was also studied by Eisenbud and Levine, where they establish general bounds on the number of hyperbolic forms that must appear in a quadratic form that is representable as a local degree. Our proof method is elementary and constructive in the case of rank 5 local degrees, while the work of Eisenbud and Levine is more general. We provide further families of examples that verify that the bounds of Eisenbud and Levine are tight in several cases.
