Séminaire d'algèbre et de géométrie du 09-04-2026
Exposé de Elad Paran
Le 09-04-2026 à 14:00, en P108 et en ligne.On Landweber's unique factorization problem
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Résumé
Let R be a unique factorization domain (UFD). In 1938 Krull asked whether the ring of formal power series R[[t]] necessarily a UFD. Krull suspected the answer is generally negative, which was confirmed by Samuel in 1961. This led to a flurry of activity surrounding the question – when is R[[t]] a UFD? While the Noetherian case is now well-understood, little is known about the non-Noetherian case. Remarkably, even the case where R = K[x1,x2,…] is the ring of polynomials in countably many variables over a field is an open problem, which was raised by Landweber in 1974. Equivalently, the problem asks whether every irreducible element in R[[t]] is prime. We outline a strategy to solve this problem, and prove partial results that were not known before: If K is of characteristic zero, then every irreducible element in R[[t]] whose constant term is square-free, or a square of a prime, is prime in R[[t]]. In this talk we shall outline the proofs of these results and discuss what remains to be done in order to solve the problem entirely. Joint work with Adam Jones (Cambridge).