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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 isto 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 adelta-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 measurein ${\Bbb R}^n$. We choose $F(y)=y^{-2}\sin^2 y$ and show that the expectationcan be approximated within relative error $0< \epsilon < 1$ in quasi-polynomialtime $(m+n)^{O(\ln (m+n)-\ln \epsilon)}$, provided each form $q_i$ depends onnot more than $r$ real variables, has common variables with at most $r-1$ otherforms and satisfies $|q_i(x)| \leq \gamma \|x\|^2/r$, where $\gamma >0$ is anabsolute constant. This allows us to distinguish between "easily solvable" and"badly unsolvable" systems in some non-trivial situations.

Testing systems of real quadratic equations for approximate solutions