Primary Decomposition in Singular

Abstract

Singular is a comprehensive and steadily growing computer algebra system, with particular emphasis on applications in algebraic geometry, commutative algebra, and singularity theory.

Singular provides highly efficient core algorithms, a multitude of advanced algorithms in the above fields, an intuitive, C-like programming language, easy ways to make it user-extendable through libraries, and a comprehensive online manual and help function.

Singular’s core algorithms handle Gröbner resp. standard bases and free resolutions, polynomial factorization, resultants, characteristic sets, and numerical root finding. Symbolic-numeric solving in Singular starts with a decomposition to a triangular system or the primary decomposition of (the radical of) an ideal. New developments for primary decomposition will be presented in this paper: identifying sub problems allows an early split of the radical. A primary decomposition of these sub problems can be lifted to (not necessary primary) decomposition of the original problem: the subsequent primary decomposition will be faster.

After the symbolic preprocessing numerical solving of these smaller and easier to solve systems can be achieved by Singular’s implementation of Laguerre’s algorithm or by integrating other systems.