Integral roots to degree $d$-forms in four variables inside a box

Question

Hi,

After the response to the following question, Rational roots to quadratic forms in 4 variables, I am now considering the following question.

Let $d \geq 2$ be a positive integer, and suppose that $F(x_1, x_2, x_3, x_4)$ is a form of degree $d$ with integer coefficients. Let $N(F,B)$ be the number of primitive integer solutions $(x_1, x_2, x_3, x_4)$ to $F(x_1, x_2, x_3, x_4) = 0$ (where primitive means that the $\gcd$ of all non-zero entries is 1, that at least one of the entries is non-zero, and that the smallest index $i$ such that $x_i \ne 0$ satisfies $x_i > 0$) satisfying $|x_i| \leq B$, for some $B > 0$. Let $\lVert F \rVert$ be the maximum of the absolute values of the coefficients of $F$.

I am wondering if a theorem of the following variety exists: In the above situation, we either have $N(F,B) \ll_\epsilon B^{\theta + \epsilon}$ for some small $\theta$, or we have $\lVert F \rVert \ll_{d, \epsilon} B^{u(d, \epsilon)}$ where $u$ is some positive function of $d$ and $\epsilon$.

As a comparison, in the three variable case we have the following result due to Bombieri and Pila, in E. Bombieri and J. Pila, "The number of integral points on arcs and ovals", Duke Mathematical Journal., 59 (1989), 337-357 which states that if $F(x,y,z)$ is a form of degree $d$, then either $N(F,B) \ll d^2$ or $\lVert F \rVert \ll B^{d(d+1)(d+2)/2}$.

The heuristic here is that if $F$ has lots of solutions in the box $|x_i| \leq B$, then its coefficients cannot be too big because then even slight variations to the coordinates would throw the value of $F$ outside the box.

Any help would be greatly appreciated.

Answer 1

There is a result of the form you want in the literature. Take a look at Theorem 4 in Heath-Brown's paper The density of rational points on curves and surfaces (Annals of Math., 155 (2002), 553–595). An easy generalisation of this gives the following. Let $G\in \mathbb{Z}[X_0,\ldots, X_n]$ be a primitive form of degree $d\geq 2$, defining a projective hypersurface $Z\subset \mathbb{P}^n$. Then either $$ \|G\|\ll_{d,n} B^{d\binom{d+n}{n}}, $$ or else there exists a form $G′\in \mathbb{Z}[X_0,\ldots, X_n]$ of degree $d$, which is not proportional to $G$, such that $G′$ vanishes at each point $x \in Z\cap \mathbb{P}^n(\mathbb{Q})$ which has height $H(x) \leq B$.

Upshot: In your case, either $\|F\|$ is small or the points you are interested actually lie on a curve in $\mathbb{P}^3$. In the latter case you get the bound $N(F,B)=O_{d}(B)$ unless the curve you produce contains a line (which must then lie in the surface $F=0$).

Answer 2

I am not sure if this answers the question, but there has been some spectacular progress in this area recently. Take a look at Pila's web page, and in particular at his paper with Wilkie http://www.maths.ox.ac.uk/system/files/PilaWilkie.pdf.

That paper includes an overview of what is known, and has references to other papers you might find useful.

Answer 3

Im not sure if this is exactly what you are looking for, but try searching google for the "dimension growth conjecture". This is essentially the conjecture that $N(X,B) \ll_{d,\varepsilon} B^{dim X + \varepsilon}$ for any projective variety $X$ of degree $d \geq 2$.

I think this is now known for any variety whose degree is not three. This follows from work of Heath-Brown, Browning, Salberger, Marmon and others. Also, Salberger has recently announced a proof for the case of degree three, however the implied constant is not uniform with respect to $X$.

These results are generally proved using a higher dimensional analogue of the determinant method of Bombieri and Pila, in particular Heath-Brown has developed a $p$-adic version of the determinant method that has proved fruitful. You might be able to use this to get the kind of result that you are looking for.

For an overview of these results I would recommend: http://www.maths.bris.ac.uk/~matdb/preprints/pila.pdf