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Let $f\colon X\to\mathbb{A}^1_k$ be a morphism from a smooth variety to anaffine line with an isolated singular point. For such a singularity, we havetwo invariants. One is a non-degenerate symmetric bilinear form (de Rham), andthe other is the vanishing cycles complex (\'etale).  In this article, we give a formula which expresses the local epsilon factorof the vanishing cycles complex in terms of the bilinear form. In particular,the sign of the local epsilon factor is determined by the discriminant of thebilinear form. This formula can be thought as a refinement of the Milnorformula, which compares the total dimension of the vanishing cycles and therank of the bilinear form.  In characteristic $2$, we find a generalization of the Arf invariant, whichcan be regarded as an invariant for non-degenerate quadratic singularities, togeneral isolated singularities.

Symmetric bilinear forms and local epsilon factors of isolated singularities in positive characteristic