The number of integer solutions to |F|<= m for decomposable form F

Number theory 1 Minute

You need to know: Polynomial in n variables, degree of a polynomial, complex numbers.

Background: Decomposable form is a polynomial F(x_1, \dots x_n)=\prod\limits_{i=1}^d (a_{i1}x_1+\dots+a_{in}x_n), where the coefficients a_{ij} are non-zero complex numbers.

The Theorem: On 24th January 2000 Jeffrey Thunder submitted to Annals of Mathematics a paper in which he proved, among other results, that for every decomposable form F of degree d in n variables with integer coefficients,  the number N_F(m) of integer solutions to the inequality |F(x_1, \dots, x_n)|\leq m is either infinite or at most c(n,d)m^{n/d}, where c(n,d) is an effectively computable constant depending only on n and d.

Short context: Counting integer solutions to equations and inequalities is an old theme in mathematics. Case d=2 of the Theorem was proved by Mahler in 1933, but progress in estimating N_F(m) for d>2 was limited. The Theorem estimates N_F(m) for all d, confirming 1989 conjecture of Schmidt.

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 1.3 of this book for an accessible description of the Theorem.

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Published by Bogdan Grechuk

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