关于$r$个对称群直积等变多项式系统的结式
Resultant of an equivariant polynomial system with respect to a direct product of $r$ symmetric groups
AI总结：
本文针对对称群直积等变多项式系统，建立了结式分解公式，并证明不变判别式可分解为若干较小结式的乘积，从而简化计算。
AI中文摘要：
本文研究了在对称群直积作用下等变的齐次多元多项式系统的结式。我们首先详细处理两个对称群乘积的情形，并建立此类系统结式的分解公式。随后我们证明，该分解及其底层组合结构可推广到$r$个对称群（有限）直积的任意情形。借助这些分解公式，我们证明了在$r$个对称群直积作用下不变的多元齐次多项式的判别式可分解为若干规模更小、更易计算的结式之积。
英文摘要：
In this paper we study the resultant of systems of homogeneous multivariate polynomials which are equivariant under the action of a direct product of symmetric groups. We first treat, in detail, the case of a product of two symmetric groups, and establish a decomposition formula for the resultant of such systems. We then show that this decomposition, together with the underlying combinatorics, extends to an arbitrary (finite) direct product of $r$ symmetric groups. Thanks to these decomposition formulas, we prove that the discriminant of a multivariate homogeneous polynomial invariant under a direct product of $r$ symmetric groups splits into a product of resultants of smaller size that are easier to compute.
