论文标题
集成二次形式的产品
Integrating products of quadratic forms
论文作者
论文摘要
我们证明，如果$ q_1，\ ldots，q_m：{\ bbb r}^n \ longrightArrow {\ bbb r} $是变量$ x_1，\ ldots的二次形式相对于标准高斯度量的$ {\ bbb r}^n $，可以在相对错误$ε> 0 $中近似近似于标准高斯量的产品$ \左（1+ q_1 \右）\ cdots \ left（1+ q_m \ right）$。 $ | q_k（x）|对于某些绝对常数$γ> 0 $和$ k = 1，\ ldots，m $，\ leqγ\ | x \ |^2 /r $。当$ q_k $解释为欧几里得空间中点配置配置的成对平方距离时，平均值可以解释为具有动感对数电势的粒子系统的分区函数。我们绘制一个可能应用的应用程序来测试实际二次方程式系统的可行性。
We prove that if $q_1, \ldots, q_m: {\Bbb R}^n \longrightarrow {\Bbb R}$ are quadratic forms in variables $x_1, \ldots, x_n$ such that each $q_k$ depends on at most $r$ variables and each $q_k$ has common variables with at most $r$ other forms, then the average value of the product $\left(1+ q_1\right) \cdots \left(1+q_m\right)$ with respect to the standard Gaussian measure in ${\Bbb R}^n$ can be approximated within relative error $ε>0$ in quasi-polynomial $n^{O(1)} m^{O(\ln m -\ln ε)}$ time, provided $|q_k(x)| \leq γ\|x\|^2 /r$ for some absolute constant $γ> 0$ and $k=1, \ldots, m$. When $q_k$ are interpreted as pairwise squared distances for configurations of points in Euclidean space, the average can be interpreted as the partition function of systems of particles with mollified logarithmic potentials. We sketch a possible application to testing the feasibility of systems of real quadratic equations.
