Three topics on Hilbert functions: Froberg's Conjecture, Hilbert coefficients and deformations of local rings

In this thesis, we present three topics concerning the study of Hilbert function. In the first topic, we obtained a partial solution to Froberg's conjecture. In the second topic, we gave the bounds for the second Hilbert coefficients of filtered modules. In the third part, we study the preservation of Hilbert function under small perturbations.