Reading the log canonical threshold of a plane curve singularity from its Newton polyhedron (Q6592712)

The log canonical threshold of a holomorphic function \(f\) on a domain of \(\mathbb{C}^n\) at a zero \(p\) of \(f\), lct\(_P(f)\), is an interesting invariant that appears in multiple contexts. It can be computed, theoretically, via, the L2 condition for holomorphic functions, the growth of the codimension of jet schemes, the poles of the motivic zeta function, the generalised and twisted Bernstein-Sato polynomial, the test ideals, the Arnold's complex oscillation index, the Hodge spectrum and the orders of vanishing on a log resolution. However few explicit computations are known.\N\NIn this paper, the author proves the following result: Assume that \(f \in \mathbb{C} \{x_1, \ldots, x_n\}\) is a nonzero power series such that \(f(\mathbf{0})=0\). Let \(c\) the unique positive rational number such that a facet \(\Delta\) of the Newton polyhedron of \(f\) contains the point \((c, \ldots, c)\). Then, lct\(_{\mathbf{0}}(f) \leq \frac{1}{c}\). Moreover, if \(n=2\) and either \(f\) is weakly normalized with respect to \(\Delta\) or the point \((c,c)\) is in the intersection of two facets, then lct\(_{\mathbf{0}}(f) = \frac{1}{c}\). This results extends a result by \textit{A. N. Varchenko} [Math. USSR, Izv. 18, 469--512 (1982; Zbl 0489.14003)].