Characteristic polynomials for symmetric forms

矩阵a的特征多项式是本科线性代数课程中经常介绍的一个基本数学对象，定义为行列式det（a - ti）。它的关键性质，即在直接和下的行为，张量积，以及对于三角矩阵它只依赖于对角项，可以用一个环同态mk ^cyc(R)—>W(R)来优雅地编码，从交换环R的循环k群，其元素由有限生成的射影R模的自同态表示，到R的Witt向量环，其元素是常系数1的R上的幂级数。Almkvist证明了这个映射是内射的，表明在某种意义上特征多项式是自同态模扩展的完全不变量。这个项目将研究这种结构在多大程度上可以扩展到对称形式。Kato构造了特征为2的域上对称形式的Witt群的完全不变量，其值为基域在其平方子域上的二次张量积。我们建议将这个不变量视为对称形式的秩或迹，并且项目的目标是将这个秩提升到一个Witt向量环，以构造Grothendieck-Witt群的不变量。我们将用于执行该程序的技术由Bökstedt， Hsiang和Madsen为Grothendieck-Witt谱提供的一个版本的环切迹图提供信息，并将自然地引导我们进一步研究真实拓扑Hochschild同调，Witt向量和de Rham-Witt复合物之间的关系。

The characteristic polynomial of a matrix A is a fundamental mathematical object which is usually introduced in an undergraduate linear algebra course, defined as the determinant det(A-tI). Its key properties, namely the behavior under direct sums, tensor products, and that for triangular matrices it depends only on the diagonal entries, can be elegantly encoded by a ring homomorphismK^cyc(R)--->W(R)from the cyclic K-group of a commutative ring R, whose elements are represented by endomorphisms of finitely generated projective R-modules, to the ring of Witt vectors of R, whose elements are power series on R with constant coefficient 1. Almkvist proves that this map is injective, showing that in a sense the characteristic polynomial is a complete invariant for endomorphisms modulo extensions.This project will investigate to which extent this construction can be extended to symmetric forms. Kato constructed a complete invariant for the Witt group of symmetric forms over a field of characteristic 2, valued in the two-fold tensor product of the base field over its subfield of squares. We propose to regard this invariant as the rank, or trace, of a symmetric form, and the goal of the project is to lift this rank to a ring of Witt vectors to construct an invariant for the Grothendieck-Witt group.The techniques we will use for carrying out this program are informed by a version of the cyclotomic trace map of Bökstedt, Hsiang and Madsen for the Grothendieck-Witt spectrum, and will naturally lead us to further investigate the relationship between real topological Hochschild homology, the Witt vectors, and the de Rham-Witt complex.