Applying Quantifier Elimination to the Birkhoff Interpolation Problem

This paper is devoted to show how to use Computer Algebra and Quantifier Elimination to solve some particular instances of the Birkhoff Interpolation Problem. In particular, this problem is completely solved for degrees less than or equal to 3 and any number of nodes and several instances of degree 4 and 5 by computing all the incidence normal poised matrices with such characteristics. The used Computer Algebra and Quantifier Elimination includes manipulation of multivariate polynomials, computation of determinants of matrices with polynomial entries and the formal manipulation of univariate polynomial inequalities by using Sturm—Habicht sequences and the Sign Determination Scheme.