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We express the Euler-Poincaré characteristic of a semi-algebraic set, which is the intersection of a non-singular complete intersection with two polynomial inequalities, in terms of the signatures of appropriate bilinear symmetric forms.

Given a closed (not necessarly compact) semi-algebraic set $X$ in ${ℝ}^n$, we construct a non-negative semi-algebraic ${𝒞}^2$ function $f$ such that $X=f^{-1}\left(0\right)$ and such that for $\delta \>0$ sufficiently small, the inclusion of $X$ in $f^{-1}\left([0,\delta ]\right)$ is a retraction. As a corollary, we obtain several formulas for the Euler characteristic of $X$.