Using methods from algebraic geometry to develop numerical approximations

Many numerical techniques are designed such that they are exact for polynomials. As a consequence, one often finds that the parameters of these techniques (e.g. collocation points and weights) solve large systems of polynomial equations. A well-known example with known solutions is Gaussian quadrature. Typically, the solution of these problems relies on highly specialised methods to simplify the equations and nonlinear numerical equation solvers. Here we will study algebraic approaches like the Groebner basis as a step in the development of the techniques to solve the polynomial equations. We will compare the new approaches with the current traditional methods.