Matrix versions of real and quaternionic nullstellensatz

A Quaternionic Nullstellensatz

We prove a Nullstellensatz for the ring of polynomial functions in n non-commuting variables over Hamilton's ring of real quaternions. We also characterize the generalized polynomial identities in n variables which hold over the quaternions, and more generally, over any division algebra.

四元数与矩阵零点定理之间的平行关系

Parallels between quaternionic and matrix nullstellensätze

本文证明了新的四元数与矩阵零点定理，并揭示了两者之间的内在联系。

A non-commutative Nullstellensatz

Let $K$ be a field and $D$ be a finite-dimensional central division algebra over $K$. We prove a variant of the Nullstellensatz for $2$-sided ideals in the ring of polynomial maps $D^n \to D$. In the case where $D = K$ is commutative, our main result reduces to the $K$-Nullstellensatz of Laksov and Adkins-Gianni-Tognoli. In the case, where $K = \mathbb R$ is the field of real numbers and $D$ is the algebra of Hamilton quaternions, it reduces to the quaternionic Nullstellensatz recently proved by Alon and Paran.

四元数上的希尔伯特零点定理

Hilbert's Nullstellensatz over quaternions

本文利用拉比诺维奇技巧证明了四元数上的零点定理，从而推广了复数域上的希尔伯特零点定理。

四元数傅里叶变换与四元数卷积的矩阵形式

On the Matrix Form of the Quaternion Fourier Transform and Quaternion Convolution

本研究探讨四元数傅里叶变换与卷积的矩阵形式，阐明其与复数域的关系，并利用理论结果提出限制四元数卷积神经网络Lipschitz常数的方法。

关于四元数的高斯-卢卡斯定理

On the Theorem of Gauss--Lucas for quaternions

该研究证明了四元数多项式导数根的分布特性，强化了高斯-卢卡斯定理的四元数版本。