Transverse slices, Ruas' conjecture, and Zariski's multiplicity conjecture for quasihomogeneous surfaces

In this work, we consider a finitely determined, quasihomogeneous, corank 1 map germ $f$ from $(\mathbb{C}^2,0)$ to $(\mathbb{C}^3,0)$. We introduce the concept of the $\mu_{\mathbf{m},\mathbf{k}}$-minimal transverse slice of $f$}. Since such a slice is a plane curve, it admits a topological normal form, which we describe explicitly. Assuming the $\mu_{\mathbf{m},\mathbf{k}}$-minimal transverse slice hypothesis, we provide a proof for the equivalence between topological triviality and Whitney equisingularity in Ruas' conjecture within this setting. We also provide a counterexample which shows that Whitney equingularity does not imply bi-Lipschitz equisingularity, given an answer to a question by Ruas. Moreover, we show that every topologically trivial $1$-parameter unfolding of $f=(f_1,f_2,f_3)$ (not necessarily with $\mu_{\mathbf{m},\mathbf{k}}$-minimal transverse slice) is of non-negative degree; that is, any additional term $\alpha$ in the deformation of $f_i$ has weighted degree not smaller than that of $f_i$. As a consequence, we provide a proof of Zariski's multiplicity conjecture for 1-parameter families of such germs.