Castles of numbers, and a bit of rethinking

Wed, 19 Nov 2025 00:00:00 +0000

In number theory, we often consider a generalization of integers called algebraic numbers. Their definition is rather elementary, but their classification is nothing but. Algebraic numbers come in families, and we can attach each family an invariant measuring its size: the castle. Kronecker proved that an algebraic integer with castle strictly less than one is zero, and that an algebraic integer with castle exactly one is a root of unity. The classification of algebraic numbers with castle less than a prescribed constant is technical, but we managed to derive it for cyclotomic integers (a subclass of algebraic numbers) with castle less than 5.01, solving a conjecture of R. M. Robinson opened in 1965.

Abstract nonsense in number theory: replacing integers by polynomials

Mon, 28 Apr 2025 00:00:00 +0000

In this short talk, we will brush a somewhat unpleasant state of affairs: for many applications, it is actually easier to use polynomials (with coefficients in a finite field) than integers.

Examples include: -Factorization: thousands of large polynomials (with coefficients in a finite field) are factored everyday in different cryptographic protocols, while factorizing a single 1024 bit number without obvious factors would take hundreds of core-years.

The Riemann hypothesis: its analogue for function fields (the field extensions that contain polynomials instead of integers), is a theorem.

Ultimately, this has to do with geometry: integers are constants, while polynomials can be seen as functions acting on a curve. We will explain how one can naturally go from numbers to polynomials, and mention my principal topic of interest: Drinfeld modules.

Other talks on Antoine Leudière

Castles of numbers, and a bit of rethinking

Abstract nonsense in number theory: replacing integers by polynomials