Simultaneous direct sum decompositions of several multivariate polynomials
Lishan Fang, Hua-Lin Huang, Lili Liao
TL;DR
The paper tackles simultaneous direct sum decompositions for sets of multivariate polynomials by extending Harrison's center theory. It defines the center $Z(f_1,\dots,f_m)$ and shows it carries a special Jordan algebra structure via $X\odot Y=\tfrac12(XY+YX)$, establishing a bijection between decomposition schemes and complete sets of orthogonal idempotents of the center. Under generic coefficients and sufficiently high degree, the center often collapses to the base field $\mathbb{k}$, making decompositions non-generic; when nontrivial idempotents exist, they induce block-diagonalizable changes of variables yielding simultaneous direct sum decompositions. The authors provide a concrete algorithm with three steps—center computation, idempotent extraction, and variable reparameterization—and illustrate the approach with several examples, highlighting both successful decompositions and limitations. This framework offers a principled, algebraic pathway to simplify families of multivariate polynomials by decoupling them into independent variable blocks, with potential applications in symbolic computation and polynomial system solving.
Abstract
We consider the problem of simultaneous direct sum decomposition of a set of multivariate polynomials. To this end, we extend Harrison's center theory for a single homogeneous polynomial to this broader setting. It is shown that the center of a set of polynomials is a special Jordan algebra, and simultaneous direct sum decompositions of the given polynomials are in bijection with complete sets of orthogonal idempotents of their center algebra. Several examples are provided to illustrate the performance of this method.