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...

Let D = 1 be a positive non-square integer and let δ = √ D or 1+ √ D 2 be a real quadratic irrational with trace t = δ + δ and norm n = δδ. Let γ = P+δ Q be a quadratic irrational for positive integers P and Q. Given a quadratic irrational γ, there exist a quadratic ideal Iγ = [Q, δ + P ] and an indefinite quadratic form Fγ(x, y) = Q(x−γy)(x−γy) of discriminant Δ = t − 4n. In the first section,...