An introduction to the determinant method: Part II

Originally, I intended this post to contain a presentation of the formal statement of the ‘main’ theorem in the $p$-adic version of the determinant method as given by Salberger. However, I find that I could not simplify his exposition significantly without losing the significance of his perspective, and so this post is now meant as an introduction of the various forms of determinant methods. Most notably, a recent paper by Dietmann and Marmon (found here) introduced a new variant of the determinant method, which prompted me to write this post.

The ‘original’ version of the determinant method was due to Enrico Bombieri and Jonathan Pila, in their seminal 1989 paper “The number of points on arcs and ovals”. Here they examined points on curves in $\mathbb{R}^2$. Most significantly, they obtained the best possible uniform result for algebraic curves of degree $d$ and so far theirs remains the only paper to address rational points on transcendental smooth curves using the determinant method (that is, a similar result has not been obtained in higher dimensions). Their argument resorts to cutting the curve into small segments, each contained in a small square, and for each such square they obtain an auxiliary curve which intersects the points on the original curve inside the square. This new curve only depends on the degree of the curve and not on the coefficients. By summing over the contributions from each small square, one obtains an upper bound for the total number of points on the original curve inside a given large square.

It was long thought that the argument of Bombieri and Pila relied  too much on plane geometry to be generalized to higher dimensions. In 2002, the game was changed by D.R. Heath-Brown in his seminal paper “The density of rational points on curves and surfaces”, published in the Annals of Mathematics in 2002, where he introduced the $p$-adic determinant method. Much of this had already been covered in the previous post, so I will not over-elaborate. Broberg and Salberger refined Heath-Brown’s arguments and placed them in proper algebraic geometric terms, allowing for easy generalization. This version of the determinant method is most effective on projective varieties.

In 2006 Heath-Brown introduced the ‘affine’ determinant method, which is still $p$-adic in nature. It can be properly seen as a generalization of his original determinant method, which only worked for projective varieties, to affine varieties. But from an algebraic geometric perspective, the arguments are actually quite different and parts of Heath-Brown’s argument remains challenging to interpret in terms of algebraic geometry. Nevertheless, this is by far the most versatile of all of the determinant methods, though it usually yields subpar results compared to other versions of the determinant method, if they apply.

In 2009 Heath-Brown introduced another variant of the determinant method, for the first time generalizing the original Bombieri-Pila argument. In particular, he showed in the paper “The sums and differences of three $k$th powers” in the Journal of Number Theory, that sometimes points on an affine surface can be interpreted as points that are in some sense ‘very close’ to an algebraic curve. This allows one to ‘save’ a dimension in the course of applying the determinant method, thereby obtaining a much sharper estimate. This method was refined and later applied to the problem of dealing with power-free values of the polynomial $x^d + c$, where $c$ is a fixed integer, also by Heath-Brown in a 2013 paper. Oscar Marmon generalized Heath-Brown’s approach to the four variable case, in the paper “The sums and differences of four $k$th powers”.

Very recently, like two months ago, Rainer Dietmann and Oscar Marmon appeared to have pioneered a new version of the determinant method. Usually, the ‘difference’ in the determinant method is how one constructs the determinant. The rows of the matrix correspond to the solutions inside a given box, while the columns represent a set of monomials. How we choose those monomials is often the key to the argument. In the projective case, we choose the monomials to have the same degree, but impose some restriction on them, say the degree of the first coordinate must be bounded. In the ‘approximate’ case, referring to the case in the previous paragraph, we choose all the monomials of a given degree. In the affine case, the monomials are chosen based on a weighted sum of the logarithms of the dimensions of the box. However, we never considered bihomogeneous monomials until now. This is what Dietmann and Marmon introduced; they considered a problem where it was natural to consider monomials which are bihomogeneous of degree $(d,e)$ say. This shows that the true potential of the determinant method is yet untapped.