On Knebusch's Norm Principle for quadratic forms over semi-local rings

Signatures of hermitian forms and the Knebusch Trace Formula

Signatures of quadratic forms have been generalized to hermitian forms over algebras with involution. In the literature this is done via Morita theory, which causes sign ambiguities in certain cases. In this paper, a hermitian version of the Knebusch Trace Formula is established and used as a main tool to resolve these ambiguities. The last page is an erratum for the published version. We inadvertently (I) gave an incorrect definition of adjoint involutions; (II) omitted dealing with the case $(H\times H, \widehat{\phantom{m}}\,)$. As $W(H\times H, \widehat{\phantom{m}}\,)= W(R\times R, \wideh…

半局部环上的二次空间与正交群

Quadratic Spaces and Orthogonal Groups over semilocal Rings

将Springer奇次度定理推广到LG环上的二次型，并建立半局部环上二次型的Scharlau与Knebusch范数原理，应用于自旋群的平坦上同调与平展范数群。

On the splitting of quasilinear $p$-forms

We study the splitting behaviour of quasilinear $p$-forms in the spirit of the theory of nondegenerate quadratic forms over fields of characteristic different from 2 using an analogue of M. Knebusch's generic splitting tower. Several new applications to the theory of quasilinear quadratic forms are given. Among them, we can mention an algebraic analogue of A. Vishik's theorem on "outer excellent connections" in the motives of quadrics, partial results towards a quasilinear analogue of N. Karpenko's theorem on the possible values of the invariant $i_1$, and a proof of a conjecture of D. Hoffman…

The norm principle for type $D_n$ groups over complete discretely valued fields

Let $K$ be a complete discretely valued field with residue field $k$ with $\mathrm{char}(k)\neq 2$. Assuming that the norm principle holds for extended Clifford groups $Ω(q)$ for every even dimensional non-degenerate quadratic form $q$ defined over any finite extension of $k$, we show that it holds for extended Clifford groups $Ω(Q)$ for every even dimensional non-degenerate quadratic form $Q$ defined over $K$.

Normal Quadratics in Ore Extensions, Quantum Planes, and Quantized Weyl Algebras

Let R be a Noetherian domain and let (σ, δ) be a quasi-derivation of R such that σ is an automorphism. There is an induced quasi-derivation on the classical quotient ring Q of R. Suppose F = t^2 - v is normal in the Ore extension R[t; σ, δ] where v ε R. We show F is prime in R[t; σ, δ] if and only if F is irreducible in Q[t; σ, δ] if and only if there does not exist w ε Q such that v = σ (w)w - δ (w). We apply this result to classify prime quadratic forms in quantum planes and quantized Weyl algebras.

调和三次齐次多项式：其Hessian范数平方是欧几里得二次型的倍数

Harmonic cubic homogeneous polynomials such that the norm-squared of the Hessian is a multiple of the Euclidean quadratic form

本文研究调和三次齐次多项式，其Hessian范数平方为欧几里得二次型的非零倍数，在所有维度构造解并在至多4维分类，提供线性共形不等价的判定技术。