Accessibility setting

Section: Application Domains

Number theory

Being able to compute quickly and reliably algebraic invariants is an invaluable aid to mathematicians: It fosters new conjectures, and often shoots down the too optimistic ones. Moreover, a large body of theoretical results in algebraic number theory has an asymptotic nature and only applies for large enough inputs; mechanised computations (preferably producing independently verifiable certificates) are often necessary to finish proofs.

For instance, many Diophantine problems reduce to a set of Thue equations of the form $P(x,y)=a$ for an irreducible, homogeneous $P\in \mathbb{Z}[x,y]$, $a\in \mathbb{Z}$, in unknown integers $x,y$. In principle, there is an algorithm to solve the latter, provided the class group and units of a rupture field of $P$ are known. Since there is no other way to prove that the full set of solutions is obtained, these algebraic invariants must be computed and certified, preferably without using the GRH.

Deeper invariants such as the Euclidean spectrum are related to more theoretical concerns, e.g., determining new examples of principal, but not norm-Euclidean number fields, but could also yield practical new algorithms: Even if a number field has class number larger than 1 (in particular, it is not norm-Euclidean), knowing the upper part of the spectrum should give a partial gcd algorithm, succeeding for almost all pairs of elements of ${𝒪}_K$. As a matter of fact, every number field which is not a complex multiplication field and whose unit group has rank strictly greater than 1 is almost norm-Euclidean  [32] , [33] .

Algorithms developed by the team are implemented in the free Pari/Gp system for number theory maintained by K. Belabas, which is a reference and the tool of choice for the worldwide number theory community.