In mathematics, the determinant method is any of a family of techniques in analytic number theory.
The name was coined by Roger Heath-Brown and comes from the fact that the center piece of the method is estimating a certain determinant. Its main application is to give an upper bound for the number of rational points of bounded height on or near algebraic varieties defined over the rational numbers. The main novelty of the determinant method is that in all incarnations, the estimates obtained are uniform with respect to the coefficients of the polynomials defining the variety and only depend on the degree and dimension of the variety.
Development
The original version of the determinant method was developed by Enrico Bombieri and Jonathan Pila in 1989. In its original context, Bombieri and Pila's results applied only to
Bombieri and Pila's result was novel because of its uniformity with respect to the polynomials defining the curves. Roger Heath-Brown obtained the analogous result of Bombieri and Pila in higher dimensions in 2002, using a different argument. Heath-Brown's approach would later be dubbed the local p-adic determinant method. The main use of Heath-Brown's determinant method has been to try to solve the so-called dimension growth conjecture.
Aside from the real-analytic approach of Bombieri and Pila and Heath-Brown's local
In 2012, this method is reformulated by the language of Arakelov theory by Huayi Chen.
In 2016, Stanley Yao Xiao obtained a generalization of Salberger's global determinant method to the setting of weighted projective space.