Integrating Products of Quadratic Forms

We prove that if $$q_1,\dots ,q_m:{\mathbb {R}}^n \rightarrow {\mathbb {R}}$$ are quadratic forms in variables $$x_1,\dots ,x_n$$ such each $$q_k$$ depends on at most r and has common with other forms, then the average value of product $$(1+q_1)\cdots (1+q_m)$$ respect to standard Gaussian measure $${\mathbb {R}}^n$$ can be approximated within relative error $$\epsilon >0$$ quasi-polynomial $$n^{O(1)}\hspace{0.55542pt}m^{O(\ln m-\ln \epsilon )}$$ time, provided $$|q_k(x)|\leqslant \gamma \hspace{0.33325pt}\Vert x\Vert ^2 /r$$ for some absolute constant $$\gamma > 0$$ $$k=1, \ldots , m$$ . The integral question is viewed as independence polynomial an auxiliary weighted graph method interpolation applied. When interpreted pairwise squared distances configurations points Euclidean space, partition function systems particles mollified logarithmic potentials. sketch possible applications testing feasibility real equations computing permanents positive definite Hermitian matrices.