Gröbner bases lie at the forefront of the algorithmic treatment of polynomial systems and ideals in symbolic computation. They are

defined as special generating sets of polynomial ideals which allow to decide the ideal membership problem via a multivariate version of

polynomial long division. Given a Gröbner basis for a polynomial ideal, a lot of geometric and algebraic information about the

polynomial ideal at hand can be extracted, such as the degree, dimension or Hilbert function.

Notably, Gröbner bases depend on two parameters: The polynomial ideal which they generate and a monomial order, i.e. a certain kind

of total order on the set of monomials of the underlying polynomial ring. Then, the geometric and ideal-theoretic information that can be

extracted from a Gröbner basis depends on the chosen monomial order. In particular, the lexicographic one allows us to solve a polynomial system.

Such a lexicographic Gröbner basis is usually computed through a change of order algorithm, for instance the seminal FGLM algorithm. In this talk,

I will present progress made to change of order algorithms: faster variants in the generic case, complexity estimates for system of critical values, computation

of colon ideals or of generic fibers.

This is based on different joint works with A. Bostan, Ch. Eder, A. Ferguson, R. Mohr, V. Neiger and M. Safey El Din.