AI 中文总结
本文针对多元函数的直和与直积分解问题，引入中心不变代数，给出分解判据与算法并应用于多元齐次多项式因式分解。
AI 中文摘要
本文研究在合适的可逆线性变量变换下，向量值多元函数能否表示为变量互不相交的向量值函数之和或积的问题。核心是针对具有二阶偏导数的多元函数集合引入的不变代数——所谓的中心。我们为满足少量分析条件的任意多元函数集合的同时加性与乘性分解提供了简单判据和算法，并将其应用于多元齐次多项式的因式分解问题，特别是那些线性形式乘积的多项式。
英文摘要
This paper addresses the problem of whether or not a vector-valued multivariate functions can be expressed as a sum or a product of vector-valued functions in disjoint sets of variables through a proper invertible linear change of variables. The crux is an invariant algebra, the so-called center, that we introduce for a set of multivariate functions with second order partial derivatives. We thus provide simple criteria and algorithms for simultaneous additive and multiplicative decompositions of any set of multivariate functions with minor analytic conditions. This is applied to the factorization problem of multivariate homogeneous polynomials, in particular those that are products of linear forms.
Comments18 pages
