Stabilized algebraic methods for multivariate polynomial root finding

We consider the problem of finding the isolated points defined by an ideal in a ring of (Laurent) polynomials with complex coefficients. Algebraic approaches for solving this use rewriting techniques modulo the ideal to reduce the problem to a univariate root finding or eigenvalue problem. We introduce a general framework for algebraic solvers in which it is possible to stabilize the computations in finite precision arithmetic. The framework is based on truncated normal forms (TNFs), which generalize Groebner and border bases. The stabilization is based on a `good' choice of basis for the quotient algebra of the ideal and on compactification of the solution space.