# Is the singular integral that come up in circle method independnet of the representatin of the equations?

Let $F_1(\mathbf{x}), F_2(\mathbf{x}) \in \mathbb{Z}[x_1, ..., x_n]$ be a degree $d$ homogeneous polynomial. For the system of equations $$F_1(\mathbf{x})= F_2(\mathbf{x}) =0,$$ we have the following singular integral $$\sigma_{\infty} = \int_{\mathbb{R}^2} I(\alpha_1, \alpha_2) d\alpha_1d\alpha_2,$$ where $$I(\alpha_1, \alpha_2) = \int_{[0,1]^n} e^{2 \pi i (F_1(\mathbf{x}) \alpha_1 + F_2(\mathbf{x}) \alpha_2 ) } dx_1...dx_n.$$

Let $G_1(\mathbf{x}) = F_1(\mathbf{x}) + g(\mathbf{x})F_2(\mathbf{x})$ and $G_2(\mathbf{x}) = F_2(\mathbf{x})$, where $g$ is some homogeneous polynomial. Then the set of solutions of $$G_1(\mathbf{x})= G_2(\mathbf{x}) =0$$ is the same as that of $F_1$ and $F_2$. Let $\sigma'_{\infty}$ be the singular integral of this system obtained by replacing $F_1$ and $F_2$ with $G_1$ and $G_2$, respectively, in the definition of singular integral above.

I suspect that $\sigma_{\infty}$ and $\sigma'_{\infty}$ are the same, since they both correspond to the same affine variety, but I don't see how I can prove this statement. Could anyone please give me a hint or explanation on how I can see this (assuming the integrals exists, etc)? Thank you very much.

• Your definitions are a bit strange: for instance, $\alpha_1$ does not appear at all in the definition of $I(\alpha_1, \alpha_2)$, and it is not obvious that the integral defining $\sigma_\infty$ is absolutely integrable (though one could perhaps still interpret this integral using some mollifier). Is there any motivation for considering this quantity $\sigma_\infty$? Commented May 12, 2016 at 17:23
• @TerryTao There was a small typo, my apologies. It is fixed, $I(\alpha_1, \alpha_2)$ does depend on $\alpha_1$ also. I believe this $\sigma_{\infty}$ is exactly what appears as the singular integral (or the real density) in many application of circle method for counting integral points on varieties. At least that was my understanding. I just took this definition from paper of Browning-Heath-Brown, "Forms in many variables and differing degrees" arxiv.org/pdf/1403.5937v2.pdf page 14 (with $M=1$ ). Commented May 12, 2016 at 18:24
• @TerryTao And in the applications I have seen one shows that for the system of equations in consideration, the integral defining $\sigma_{\infty}$ is absolutely convergent. Thanks! Commented May 12, 2016 at 18:28

Formally (ignoring issues of integrability), one has (writing $e(\theta) := e^{2\pi i\theta}$)

$$\sigma'_\infty = \int_{{\bf R}^2} \int_{[0,1]^n} e( F_1({\bf x}) \alpha_1 + g({\bf x}) F_2({\bf x}) \alpha_1 + F_2({\bf x}) \alpha_2)\ d{\bf x} d\alpha_1 d\alpha_2$$ $$= \int_{{\bf R}} \int_{[0,1]^n} e( F_1({\bf x}) \alpha_1) ( \int_{\bf R} e( (\alpha_2 + g({\bf x}) \alpha_1) F_2({\bf x}))\ d\alpha_2 ) d{\bf x} d\alpha_1.$$ Similarly for $\sigma_\infty$ without the $g({\bf x})$ term. If one formally translates $\alpha_2$ by $g({\bf x}) \alpha_1$ in the inner integral, one obtains the desired identity $\sigma_\infty = \sigma'_\infty$.

To make this rigorous, one has to damp the ${\bf R}^2$ integral by some mollifier (e.g. a gaussian $e^{-\varepsilon_1 \alpha_1^2 - \varepsilon_2 \alpha_2^2}$, and it will be convenient to make $\varepsilon_2$ a bit smaller than $\varepsilon_1$ so that the aforementioned translation has negligible impact on the mollifier) in order to justify the various changes of variable, and use suitable decay bounds on oscillatory integrals (related to whatever bounds were needed to ensure the absolute integrability of $I$) to take limits properly.

From the point of view of the theory of distributions, one morally has $$\sigma_\infty = \int_{[0,1]^n} \delta(F_1({\bf x})) \delta(F_2({\bf x}))\ d{\bf x}$$ where $\delta$ is the Dirac delta, which makes the claimed invariance formally obvious. If the variety $\Sigma := \{ {\bf x} \in [0,1]^n: F_1({\bf x}) = F_2({\bf x}) = 0 \}$ is smooth, this expression can also be written as $$\int_{\Sigma} \frac{dS}{|dF_1 \wedge dF_2|},$$ where $dS$ is surface measure, which is another way to make the claimed invariance manifest.

• Thank you very much for this answer! I am just going through the details to make sure I can do it and I think it will work out. Greatly appreciated! Commented May 12, 2016 at 23:17
• The exact condition I have for $I(\alpha_1, \alpha_2)$ , for both the $F$'s and for the $G$'s, is that $I(\alpha_1, \alpha_2) \ll \min \{ 1, | \boldsymbol{\alpha} |^{-3} \}$, where $| \boldsymbol{\alpha} |$ is the sup norm. Do you think this is enough to make your first argument work? (I thought I had it following your argument, but I realized an error with what I did, so I was wondering if this condition was sufficient) Commented May 28, 2016 at 15:30

The comparison between singular integrals and “canonical” volumes has been explained in

Emmanuel Peyre, MR 1340296 Hauteurs et mesures de Tamagawa sur les variétés de Fano, Duke Math. J. 79 (1995), no. 1, 101--218.

in the framework of Manin's conjecture for Fano varieties.

The argument is similar in spirit to the one explained by Terry Tao in his answer, and works for any complete intersection.

I have developed it further in a paper with Yuri Tschinkel:

Antoine Chambert-Loir and Yuri Tschinkel, MR 2740045 Igusa integrals and volume asymptotics in analytic and adelic geometry, Confluentes Math. 2 (2010), no. 3, 351--429.

• I was wondering does one of the papers answer my question? If so would it be possible to point out the page number? Thank you very much! Commented May 29, 2016 at 0:40