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的等维子集，即每个子集的不可约分支具有相同维数。该算法的核心是将三角集理论（亦称正则链）的思想与Gröbner基相结合，用于编码和处理局部闭代数集。基于此，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见的某些一般性假设（例如关于Noether位置的假设）。这使得算法能够在更具结构性的系统上生成精细分解，而确保一般性假设往往会破坏这些系统的结构。实际实验表明，与现有最优实现相比，该算法具有更高的效率。