We describe a recursive algorithm that decomposes an algebraic set into locally closed equidimensional sets, i.e. sets which each have irreducible components of the same dimension. At the core of this algorithm, we combine ideas from the theory of triangular sets, a.k.a. regular chains, with Gr\"obner bases to encode and work with locally closed algebraic sets. Equipped with this, our algorithm avoids projections of the algebraic sets that are decomposed and certain genericity assumptions frequently made when decomposing polynomial systems, such as assumptions about Noether position. This makes it produce fine decompositions on more structured systems where ensuring genericity assumptions often destroys the structure of the system at hand. Practical experiments demonstrate its efficiency compared to state-of-the-art implementations.

翻译：本文描述了一种递归算法，该算法将代数集分解为局部闭等维集，即每个集合的不可约分支具有相同维数。该算法的核心结合了三角集理论（也称为正则链）与Gröbner基的思想，以编码和处理局部闭代数集。借助这一方法，我们的算法避免了被分解代数集的投影以及多项式系统分解中常做的某些一般性假设（如Noether位置假设）。这使得算法能够对更具结构化的系统产生精细分解，而确保一般性假设往往会破坏系统的固有结构。实际实验证明了其相较于现有实现方法的效率优势。