On two inequality conjectures for the k-th Yau numbers of isolated hypersurface singularities

Let (V, 0) be an isolated hypersurface singularity defined by the holomorphic function f: (Cⁿ, 0) → (C, 0). In our previous work, we introduced a series of novel Lie algebras associated to (V, 0), i.e., k-th Yau algebra L^k(V) , k≥ 0. It was defined to be the Lie algebra of derivations of the k-th moduli algebras A^k(V) = O_n/ (f, m^kJ(f)) , k≥ 0 , where m is the maximal ideal of O_n. I.e., L^k(V) : = Der (A^k(V) , A^k(V)). The dimension of L^k(V) was denoted by λ^k(V). The number λ^k(V) , which was called k-th Yau number, is a subtle numerical analytic invariant of (V, 0). Furthermore, we formulated two conjectures for these k-th Yau number invariants: a sharp upper estimate conjecture of λ^k(V) for weighted homogeneous isolated hypersurface singularities (see Conjecture 1.2) and an inequality conjecture λ^(k+1)(V) > λ^k(V) , k≥ 0 (see Conjecture 1.1). In this article, we verify these two conjectures when k is small for large class of singularities.