The notion of Hilbert function plays a central role in commutative algebra, in algebraic geometry and in computational algebra. The Hilbert function of a local (or graded) ring is a polynomial function and the coefficients of the corresponding polynomial, called Hilbert coefficients, may capture several numerical and homological invariants of the ring itself. Starting from classical results of S. Abhyankar, D. Northcott and P. Samuel, many papers have been written in this field which is considered an important part of the theory of blowing-up rings.
In this talk we present some techniques and we will focus on some open problems which are motivating the recent research on the topic.