---
abstract: 'We give a generalization of the duality of a zero-dimensional complete intersection to the case of one-dimensional almost complete intersections, which results in a [*Gorenstein module*]{} $M=I/J$. In the real case the resulting pairing has a signature, which we show to be constant under flat deformations. In the special case of a non-isolated real hypersurface singularity $f$ with a one-dimensional critical locus, we relate the signature on the jacobian module $I/J_f$ to the Euler characteristic of the positive and negative Milnor fibre, generalising the result for isolated critical points. An application to real curves in ${{\mathbb P}}^2({{\mathbb R}})$ of even degree is given.'
author:
- Duco van Straten and Thorsten Warmt
bibliography:
- 'math.bib'
title: 'Gorenstein-duality for one-dimensional almost complete intersections – with an application to non-isolated real singularities'
---

Introduction
============

The algebraic determination of the number of real roots of a polynomial has a long history going back at least to Descartes. Of particular relevance are the methods of Sylvester and Hermite that determine the number of real roots as the [*signature of an associated quadratic form*]{}. For a nice account of the classical approaches we refer to [@weber].

In a similar spirit, the celebrated theorem 