Counting integer points on affine surfaces with a side condition

Counting integer points on affine surfaces with a side condition, Discrete Analysis 2025:12, 25 pp.

This paper is concerned with special instances of the general problem of bounding the number of solutions to Diophantine equations with the variables constrained to lie in a large box. In 1989 Bombieri and Pila developed a general method to obtain such bounds. For example if $f \in {\mathbb R}[x,y]$ is an irreducible polynomial of degree $d$ then they established that the number of integer points on the curve $f(x,y)=0$ with $|x|$, $|y| \le B$ is at most $O(B^{\frac 1d +\epsilon})$. One key feature of this result is that the bound on the number of solutions does not depend on the curve $f(x,y)=0$. By contrast, smooth algebraic curves of genus at least $1$ would have only finitely many integer points (by Siegel’s theorem), but this bound would certainly depend on the choice of the curve. The method pioneered by Bombieri and Pila has come to be known as the determinant method, and its scope has been extended considerably with notable developments due to Heath-Brown and Salberger. The original Bombieri–Pila result may be thought of as a real analytic determinant method (applying to transcendental curves as well); Heath-Brown developed a powerful $p$-adic analogue of the method, and Salberger a “semi global” version of the method.

Let $f \in {\mathbb Z}[x, y, z]$ be an absolutely irreducible polynomial of degree $d$, and consider the number $N(f;B)$ of zeros $(x,y,z)$ of $f$ with $|x|$, $|y|$, $|z| \le B$. As noted by Pila, the original Bombieri–Pila method could be used to show that $N(f;B) = O(B^{1+1/d+\epsilon})$, with the implied constant allowed to depend on $d$ and $\epsilon$, but independent of the polynomial $f$. Salberger’s refined version permits the better bound $O(B^{2/\sqrt{d}+\epsilon} + B^{1+\epsilon})$. Browning and Verzobio’s paper considers the problem of bounding the number of solutions to $f(x,y,z)=0$ in a box, subject to the additional condition that $g(x,y,z) \equiv 0 \bmod q$ for another polynomial $g \in {\mathbb Z}[x,y,z]$ and a natural number $q$. The goal is now to obtain better bounds for the number of solutions satisfying this side condition, especially when $q$ is large.

Drawing on Salberger’s version of the determinant method, this paper establishes a general result on this problem, and the flavor of the result is illustrated through two sample applications. The first application asks for a count of solutions to $ax^2 + by^2 +cz^2 =n$ with $|x|$, $|y|$, $|z| \le B$, assuming that $c$ has the largest size among $a$, $b$, $c$. Here one takes $f(x,y,z)=ax^2+by^2+cz^2-n$, and the side condition is that $g(x,y) = ax^2+by^2 - n \equiv 0 \bmod c$. An easy estimate for the number of solutions is $O(B^{1+\epsilon})$, and the methods of the paper produce (under natural assumptions on $a$, $b$, $c$, $n$) an improved bound $O(B^{\frac 76+\epsilon}/|c|^{\frac 16} + B^{\frac 12 +\epsilon})$.

The second application counts solutions to $x^k + y^l + z^m + w^k = N$, where $k\ge 13$ is an odd natural number and $k > l > m \ge 2$. Since $k$ is odd, $x^k + w^k$ is divisible by $q= (x+w)$, and the problem can be placed in the framework of the paper with the side condition being $y^l + z^m - N \equiv 0 \bmod q$.