Real solution isolation with multiplicity of zero-dimensional triangular systems

Existing algorithms for isolating real solutions of zero-dimensional polynomial systems do not compute the multiplicities of the solutions. In this paper, we define in a natural way the nmltiplicity of solutions of zero-dimensional triangular polynomial systems and prove that our definition is equivalent to the classical definition of local （intersection） multiplicity. Then we present an effective and complete algorithm for isolat- ing real solutions with multiplicities of zero-dimensional triangular polynomial systems using our definition, The algorithm is based on interval arithmetic and square-free factorization of polynomials with real algebraic coefficients. The computational results on some examples from the literature are presented.