On Milnor and Tjurina numbers of Pairs of Holomorphic Functions Germs

Motivated by the bifurcation formula for the Milnor number of pairs of holomorphic function germs, we introduce, via foliation theory, the Tjurina number of such pairs.

We prove that this is an analytic invariant, derive an explicit formula for it, and establish several of its properties, including a bifurcation formula.

As an application, we characterize semitame meromorphic function germs in terms of this invariant.

We also investigate several properties of the Milnor number of a pair of holomorphic function germs.

In particular, we establish a version of Teissier's Lemma for pairs, derive an upper bound for the Milnor number of a pair, and apply these results to generic pencils of algebraic curves.