An introduction to the determinant method: Part I

I feel like I should devote a section of this blog to my own research activities. My main area of interest currently is the so-called determinant method, first developed by Enrico Bombieri and Jonathan Pila in 1989 to investigate rational points on plane curves, and extended greatly by D.R. Heath-Brown in 2002 to include higher dimensional varieties. The approach of Bombieri and Pila greatly differ from Heath-Brown’s, but they are still connected in essence. Heath-Brown’s method was then properly placed by Broberg and subsequently Salberger in a proper algebraic setting. In particular, they showed that the somewhat ad-hoc construction of Heath-Brown can be replaced by a construction compatible to considering a polynomial ideal and a suitable monomial ordering.

Essentially, the basic object of study is an algebraic variety $V$, either an affine variety in $\mathbb{A}^n$ or $\mathbb{P}^n$. Usually we are only concerned with rational or integral points, though there is little issue with passing this method to number fields. A standard technique is to consider plane sections of $V$ and count the points on each plane section. This has produced good results, but it would be profitable to consider intersections with hypersurfaces of larger degree than just hyperplanes. This is what the determinant method is able to provide: the output of the determinant method is a relatively small set of hypersurfaces of bounded degree that counts all of the points of $V$ in a box.

Remark: In more recent literature the notion of “relatively few hypersurfaces of relatively low degree” has been replaced with “a hypersurface of relatively low degree”. The two notions are equivalent; the second obtained from the first by taking the product of the polynomials defining the hypersurfaces in the first case.

In this post we will mainly be concerned with determinant methods of Heath-Brown type, which is to say that we will work with $p$-adic norms. In Heath-Brown’s original paper, only the hypersurface case was dealt with, so we will start there. Suppose that $X$ is a hypersurface in $\mathbb{P}^{n-1}$ generated by a form $F(x_1, \cdots, x_n) \in \mathbb{Z}[x_1, \cdots, x_n]$. The idea here is that we will collect all solutions inside a box that are congruent to each other modulo some auxiliary prime $p$ together. Let $X_p$ denote the reduction of the variety $X$ modulo $p$, and let $P$ be a $\mathbb{F}_p$ point on $X_p$. Now suppose that there are $R$ of points $(\xi_1^{(j)}, \cdots, \xi_n^{(j)}), 1 \leq j \leq R$ points on $X$ such that $|\xi_m^{(j)}| \leq B_m$ for $j = 1, \cdots, R$ and $1 \leq m \leq n$ where $B_1, \cdots, B_n$ are positive numbers, which reduce to $P$ modulo $p$. Now choose a set of $s$ monomials, evaluate them at each of the $R$ points, and then consider the resulting $s \times R$ matrix $M$. The goal is to show that the rank of $M$ is at most $s-1$, as then we will be able to find a non-zero polynomial which vanishes at all $R$ points. More formally, let us write $f_1, \cdots, f_s$ for the monomials. If $R < s$ then it is trivial that $M = (f_i(\xi^{(j)}))$ has rank at most $s-1$, so we assume that $R \geq s$ and consider a $s \times s$ minor. The goal then is to show that this square matrix has determinant zero (this is why the method is called the determinant method), while ensuring that the output polynomial is coprime to $F$ (otherwise the intersection is not a variety of lower dimension but a component, which means we fare no better than the original situation). To do this, Heath-Brown showed that the resulting determinant $\Delta$ is divisible by a large power of the prime $p$, and by choosing the power sufficiently large one can show that this power exceeds the upper bound for the determinant $\Delta$ which depends on the size of the box.

Before diving into the proofs of Heath-Brown’s famous “theorem 14” in his paper On the density of rational points on curves and surfaces, published in the Annals of Mathematics in 2002, we give some consequences of the determinant method. First, the greatest technical flexibility introduced by the method is that it is uniform with respect to the polynomial $F$ and instead only depend on the degree $d$ and the dimension $n$. This in turns afford great flexibility in inductive arguments. Perhaps the greatest achievement of the determinant method to date is the proof of the so-called dimension growth conjecture, now a theorem due to joint work of Browning, Heath-Brown, and Salberger, which states

Theorem: (Browning, Heath-Brown, Salberger) Let $X \subset \mathbb{P}^{n}$ be an integral projective variety of dimension $r$, and let $N(X; B)$ denote the set of rational points on $X$ of height at most $B$. Then we have

$\displaystyle N(X; B) = O_{d, \epsilon}\left(B^{r + \epsilon}\right).$

Note that this bound is uniform with respect to $X$, as the implied constant only depends on $d$ and $\epsilon$.

The determinant method has found specific application to many diophantine problems. For instance, Heath-Brown used the determinant method to investigate the sums of like powers, and more generally, equal sums of binary forms. In particular, he investigated the two (similar and related) equations:

$\displaystyle x_1^d + x_2^d = x_3^d + x_4^d$

and

$\displaystyle F(x_1, x_2) = F(x_3, x_4)$

where $F \in \mathbb{Z}[x,y]$ is a binary form. Another class of diophantine problems that have found application with the determinant method, which is the focus of my research, is the set of points for which a given polynomial of either a single variable or two variables takes on power-free values. We separate the list of results into the single variable and two variables cases. Here, $f(x)$ or $f(x,y)$ will be assumed to have integer coefficients, irreducible, and have no fixed $k$-th power divisor. $d$ will denote the degree of $f$. Then we have the following consequences of the determinant method:

Theorem: (Heath-Brown, 2006) If $k \geq (3d+2)/4$, then $f(x)$ will take on $k$-free values for a positive proportion of integers $x \in \mathbb{Z}$. The same is true for prime inputs.

Theorem: (Browning, 2011) If $k \geq (3d+1)/4$, then $f(x)$ will take on $k$-free values for a positive proportion of integers. The same is true for prime inputs.

Theorem: (Heath-Brown, 2011) If $f(x) = x^d + c$, where $c \in \mathbb{N}$ is a square-free number, and if $k \geq (5d+2)/9$, then $f(x)$ will take on $k$-free values for a positive proportion of integers. The same is true for prime inputs.

We note that the statement “same is true for prime inputs” is not trivial. Indeed, Erdős proved a version of the first two theorems in the 1950s with the condition $k = d-1$, and conjectured that the result should hold for prime inputs as well. The prime input version of Erdős’s theorem was not proved until 2006 by Harald Halfgott. The determinant method proof of the above theorems handles the case of prime inputs with no additional effort.

For the two-variable case, we have the following two theorems of Browning.

Theorem: (Browning, 2011) If $f(x,y)$ is a general polynomial satisfying the hypotheses above, then $f$ takes on $k$-free values for a positive portion of elements in $\mathbb{Z}^2$ as soon as $k > 39d/64$.

Theorem: (Browning, 2011) If we further assume that $f(x,y)$ is homogeneous, then we can reduce the lower bound to $k > 7d/16$.

The reason for the improvement in the homogeneous case is due to a much better estimate that is not conditioned on any relationship between $k$ and $d$, due to George Greaves, which is only available when $f(x,y)$ splits completely over $\overline{\mathbb{Q}}$ (this was observed much later by Hooley; Greaves’ original paper only involved the homogeneous case where this condition was trivial).

I have been able to make the following improvement to Browning’s second theorem above

Theorem: (X, 2013) If $f(x,y)$ is a binary form satisfying the conditions listed above, then it suffices for $k > 7d/18$.