Zero-dimensional families of polynomial systems

  • Lorenzo Robbiano
  • Maria-Laura Torrente
Keywords: Algebraic family, Real zeros, Condition number


If a real world problem is modelled with a system of polynomial equations, the coefficients are in general not exact. The consequence is that small perturbations of the coefficients may lead to big changes of the solutions. In this paper we address the following question: how do the zeros change when the coefficients of the polynomials are perturbed? In the first part we show how to construct semi-algebraic sets in the parameter space over which the family of all ideals shares the number of isolated real zeros. In the second part we show how to modify the equations and get new ones which generate the same ideal, but whose real zeros are more stable
with respect to perturbations of the coefficients.