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 multiplicity 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 isolating 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.