On the Jacobian polygon and Łojasiewicz exponent of isolated complex hypersurface singularities

Given an isolated hypersurface singularity ( X , 0 ) ⊂ ( C n + 1 , 0 ) defined by a holomorphic function f : ( C n + 1 , 0 ) → ( C , 0 ) , we introduce an alternating version of Teissier's Jacobian Newton polygon, and prove formulas for both these invariants in terms of an embedded resolution of ( X , 0 ) . The formula for the alternating version has an advantage, in that for Newton nondegenerate functions, it can be calculated in terms of volumes of faces of the Newton diagram, whereas a similar formula for the original non-alternating version includes mixed volumes. The Milnor fiber can be given a handlebody decomposition, with handles corresponding to intersection points with the polar curve in generic plane sections of the singularity. This way we obtain a Morse–Smale complex. Teissier associates with each branch of the polar curve a vanishing rate, and we show that this induces a filtration of the Morse–Smale complex. We apply this result in order to calculate the Łojasiewicz exponent in terms of the alternating Jacobian polygon, but we expect it to be of further independent interest. In the case of a Newton nondegenerate hypersurface, our result produces a formula for the Łojasiewicz exponent in terms of Newton numbers of certain subdiagrams. This statement is related to a conjecture by Brzostowski, Krasiński and Oleksik, for which we provide a counterexample. Our formula for the Łojasiewicz exponent is based on a global calculation over the Newton diagram, rather than locally specifying a subset of the facets to consider, as in this conjecture. We conjecture a similar statement, which is based on our formula and inspired by the nonnegativity of local h -vectors.