Student Commutative Algebra Seminar

The Briançon-Skoda theorem relates the powers on an ideal with their integral closures. In this talk, we'll discuss this theorem in the simplest, yet interesting case of a polynomial ring over a field. For example, it implies the following non-trivial fact about polynomials in two variables: for any three polynomials f, g and h in two variables, the product f^2 g^2h^2 is in the ideal generated by f^3, g^3 and h^3, We'll see a proof of this theorem in positive characteristics using the operation of tight closure. Time permitting, we'll discuss how to use this to deduce the theorem in characteristic zero. We'll introduce the relevant definitions including that of integral closure, tight closure, regular rings etc. This talk will be accessible to anyone taking Math 614. Speaker(s): Swaraj Pande (UM)