  Consider systems of equations $q_i(x)=0$, where $q_i: {\Bbb R}^n
\longrightarrow {\Bbb R}$, $i=1, \ldots, m$, are quadratic forms. Our goal is
to tell efficiently systems with many non-trivial solutions or near-solutions
$x \ne 0$ from systems that are far from having a solution. For that, we pick a
delta-shaped penalty function $F: {\Bbb R} \longrightarrow [0, 1]$ with
$F(0)=1$ and $F(y) < 1$ for $y \ne 0$ and compute the expectation of $F(q_1(x))
\cdots F(q_m(x))$ for a random $x$ sampled from the standard Gaussian measure
in ${\Bbb R}^n$. We choose $F(y)=y^{-2}\sin^2 y$ and show that the expectation
can be approximated within relative error $0< \epsilon < 1$ in quasi-polynomial
time $(m+n)^{O(\ln (m+n)-\ln \epsilon)}$, provided each form $q_i$ depends on
not more than $r$ real variables, has common variables with at most $r-1$ other
forms and satisfies $|q_i(x)| \leq \gamma \|x\|^2/r$, where $\gamma >0$ is an
absolute constant. This allows us to distinguish between "easily solvable" and
"badly unsolvable" systems in some non-trivial situations.
