# Singularity structure of integrals of rational functions

Suppose I have a convergent integral of the form $\int_0^1dx_1\dots\int_0^1 dx_n \frac{P(x_i)}{Q(x_i)}$, where $P$ and $Q$ are polynomial functions of $n$ nonnegative real variables $x_i$. Let the coefficients of the various monomials in $Q$ be $a_k$. Depending on the structure of $Q$, the integral could develop singularities in the limit that some of the coefficients $a_k$ vanish. For example, the integral $\int_0^1 dx\frac 1 {a+x}$ goes like $-\log a$ as $a\to 0$. Is there a systematic way to extract the structure of these singularities for general multidimensional integrals?

The particular example I'm interested in is the following: I'd like to determine the singularities of

$\begin{eqnarray} \int_0^1 d\lambda_1\int_0^1 d\lambda_2 \int_0^1d\alpha_1 \int_0^1d\alpha_2 \frac{\alpha_1\alpha_2}{(\lambda_2\alpha_1\alpha_2+\lambda_1\alpha_1\alpha_2+a\alpha_2\lambda_1^2+b\alpha_1\lambda_2^2+c\alpha_1\alpha_2)^2}, \end{eqnarray}$

as $a,b,$ and $c$ tend to zero. If necessary, it's alright to assume they all tend to zero at the same rate. I naively expect a leading singularity that looks like $\log a \log b \log c$, followed by sub-leading singularities that look like $\log a \log b$ times some convergent integral, $\log b \log c$ times some other convergent integral, etc.. For my purposes, these $\log$-squared terms suffice, but I'd be interested to know how to systematically go further.

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