On paths in the \(\mu\)-constant and \(\mu^\ast\)-constant strata (Q6614433)

This article investigates the connection between polynomial function-germs in \( \mathbb{C}^n \) that share an isolated singularity at the origin, specifically focusing on the Milnor number \(\mu \) and Teissier's \( \mu^* \)-sequence invariants. The authors demonstrate that if two such function-germs \( f \) and \( f' \) can be linked by a continuous path within the \( \mu \)-constant (or \( \mu^* \)-constant) stratum, they can also be joined by a piecewise complex-analytic path defined naturally in the same stratum.\N\NThe paper introduces two constructible sets, \( W(n,m, \mu) \) and \( W^*(n,m, \mu^*) \), and explores their properties as key tools to achieve the main results. The authors use these constructs to prove the main theorem: any two polynomial function-germs in the same path-connected component of the \( \mu \)-constant (or \( \mu^* \)-constant) stratum can be connected via a piecewise complex-analytic path. This result implies that certain topological invariants, such as the monodromy zeta function and the local embedded topology of the link, remain consistent across such paths.\N\NThe paper also addresses the algebraic and topological structure of these constructible sets, leveraging results from algebraic geometry to support the proofs. The findings are expected to be useful for further exploration of invariants in singularity theory and geometric analysis.