The Hasse principle holds for pairs of diagonal cubic forms in s>=13 variables

Number theory 1 Minute

You need to know: p-adic fields {\mathbb Q}_p.

Background: If a system of equations has a solution with all variables 0, we call such solution trivial and all other solutions non-trivial. We say that a collection of systems of equations satisfies the Hasse principle if, whenever one of the systems has a non-trivial solution in real numbers and in all the p-adic fields {\mathbb Q}_p for all primes p, then it has a non-trivial solution in rational numbers.

The Theorem: On 14th April 2005, Jörg Brüdern and Trevor Wooley submitted to the Annals of Mathematics a paper in which they proved that for any integer s\geq 13, and any integer coefficients a_j, b_j, \, 1\leq j \leq s, the system of equations \sum\limits_{i=1}^sa_ix_i^3 = \sum\limits_{i=1}^sb_ix_i^3 = 0 has a non-trivial solution in integers if and only if it has a non-trivial solution in the 7-adic field.

Short context: The Hasse principle provides necessary and sufficient conditions for the existence of non-trivial integer solutions in a system of equations. The famous Hasse–Minkowski theorem states that this principle holds for any quadratic equation in s variables in the form \sum\limits_{i=1}^s \sum\limits_{j=1}^s a_{ij}x_ix_j=0 with integer coefficients a_{ij}. For cubic equations, the Hasse principle fails in general, but The Theorem implies that it holds for the system \sum\limits_{i=1}^sa_ix_i^3 = \sum\limits_{i=1}^sb_ix_i^3 = 0 in s\geq 13 variables. This result was earlier known for s\geq 14, and fails for s\leq 12, hence the condition s\geq 13 in the Theorem is the best possible.

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 7.9 of this book for an accessible description of the Theorem.

Go to the list of all theorems

Unknown's avatar

Published by Bogdan Grechuk

Published

Leave a comment