Derivations of generalized local algebras and invariants of isolated hypersurface singularities

Let ( 𝑊 , 0 ) ⊂ ( ℂ 𝑛 , 0 ) be an isolated hypersurface singularity defined by a holomorphic germ 𝑔 : ( ℂ 𝑛 , 0 ) → ( ℂ , 0 ) . For 𝑖 ≥ 1 , let J a c 𝑖 ⁢ ( 𝑔 ) be the ideal generated by all i -th order partial derivatives of g , and for 𝑝 ≥ 1 set 𝐽 𝑝 ⁡ ( 𝑔 ) : = ∑ 𝑝 𝑖 = 1 J a c 𝑖 ⁢ ( 𝑔 ) . For 𝑘 ≥ 0 , we consider the generalized local algebra 𝑀 𝑘 , 𝑝 ⁡ ( 𝑊 ) : = 𝒪 𝑛 / ( 〈 𝑔 〉 + 𝔪 𝑘 ⁢ 𝐽 𝑝 ⁡ ( 𝑔 ) ) and its derivation Lie algebra 𝐿 𝑘 , 𝑝 ⁡ ( 𝑊 ) : = 𝐷 𝑒 𝑟 ⁡ ( 𝑀 𝑘 , 𝑝 ⁡ ( 𝑊 ) , 𝑀 𝑘 , 𝑝 ⁡ ( 𝑊 ) ) . The dimension 𝜆 𝑘 , 𝑝 ⁡ ( 𝑊 ) : = d i m ℂ ⁡ 𝐿 𝑘 , 𝑝 ⁡ ( 𝑊 ) is a numerical analytic invariant of ( 𝑊 , 0 ) ; hence the numbers 𝜆 𝑘 , 𝑝 ⁡ ( 𝑊 ) form a family of analytic invariants extending the classical Yau number, which is recovered when ( 𝑘 , 𝑝 ) = ( 0 , 1 ) . Focusing on the case ( 𝑘 , 𝑝 ) = ( 1 , 2 ) , we compute 𝜆 1 , 2 ⁡ ( 𝑊 ) for the binomial and trinomial weighted homogeneous isolated singularities. We also formulate the sharp upper estimates conjecture for 𝜆 𝑘 , 𝑝 ⁡ ( 𝑊 ) and prove it for a large class of singularities.