Solving zero-dimensional algebraic systems
Let \(X_ 1,\ldots,X_ n\) be \(n\) variables ordered as \(X_ 1<X_ 2<\cdots<X_ n\). The main variable of a polynomial is the greatest variable in it. A set of \(n\) polynomials is called \textit{triangular} if the main variable of the \(i\)-th polynomial is \(X_ i\) and monic in \(X_ i\). Using this terminology, the author first remarks: Proposition 1. If \(K\) is a field, any maximal ideal in \(K[X_ 1,\ldots,X_ n]\) has a triangular system of generators. Proposition 2. Every system with a finite number of solutions (in an algebraic closure of \(K\)) is equivalent to the union of a finite number of triangular systems. Then he shows how to compute numerically the common zeros of a triangular set.
scientific article; zbMATH DE number 56190