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Abstract

Given a hypersurface singularity $(X,0) \subset (\mathbb{C}^{n+1},0)$ defined by a holomorphic function $f:(\mathbb{C}^{n+1},0) \to (\mathbb{C},0)$, we introduce an alternating version of Teissier's Jacobian Newton polygon, associated with a complex isolated hypersurface singularity, 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 nonalternating 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 {\L}ojasiewicz 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 {\L}ojasiewicz exponent in terms of Newton numbers of certain subdiagrams. This statement is related to a conjecture by Brzostowski, Krasi\'nski and Oleksik, for which we provide a counterexample. Our formula for the {\L}ojasiewicz 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.