Squarefree normal representation of zeros of zero-dimensional polynomial systems

For any zero-dimensional polynomial ideal $I$ and any nonzero polynomial F, this paper shows that the union of the multi-set of zeros of the ideal sum $I+〈F〉$ and that of the ideal quotient $I:〈F〉$ is equal to the multi-set of zeros of $I$, where zeros are counted with multiplicities. Based on this zero relation and the computation of Gröbner bases, a complete multiplicity-preserved algorithm is proposed to decompose any zero-dimensional polynomial set into finitely many squarefree normal triangular sets, resulting in a squarefree normal representation for the zeros of the polynomial set. In the representation the multiplicities of the zeros of the triangular sets can be read out directly. Examples and experiments are presented to illustrate the algorithm and its performance.