Testing systems of real quadratic equations for approximate solutions

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 $x\mathrm{\ne }0$ from systems that are far from having a solution. For that, we pick a delta-shaped penalty function $F:\mathbb{R}⟶[0,1]$ with $F(0)=1$ and $F(y)<1$ for $y\mathrm{\ne }0$ and compute the expectation of $F(q_1(x)) \cdots F(q_m(x))$forarandom$x$ sampled from the standard Gaussian measure in ${\mathbb{R}}^n$. We choose $F(y)=y^{-2}{\sin }^{2}y$ and show that the expectation can be approximated within relative error $0<ϵ<1$ in quasi-polynomial time $(m+n{)}^{O(\ln (m+n)-\ln ϵ)}$, 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 $\mathrm{\mid }{q}_i(x)\mathrm{\mid }\le \gamma \mathrm{\parallel }x{\mathrm{\parallel }}^2\mathrm{/}r$, where $\gamma >0$ is an absolute constant. This allows us to distinguish between “easily solvable” and “badly unsolvable” systems in some non-trivial situations.