Three Conjectures on the Łojasiewicz Exponent of an Isolated Hypersurface Singularity

For an isolated hypersurface singularity f:(Cn,0)→(C,0), we study the Łojasiewicz exponent ł(f) of f. We formulate three conjectures concerning: (i) attainment of ł(f) on coordinate polar curves, (ii) topological invariance of ł(f), and (iii) a local Newton-diagram formula for ł(f) in the Kushnirenko non-degenerate case. We compute ł(f) for all singularities of modality at most 2 in Arnold’s list and for several further families, obtaining systematic evidence in favor of these conjectures. We also find to what extent the collection of numerical invariants {modality(f),ł(f),μ(f),corank(f),stabord(f)} determines the stable equivalence class of f for the aforementioned singularities.