Hilbert function of local cohomology and Presburger arithmetic

Local cohomology was introduced by Grothendieck in early 1960s. Since then it has become an essential tool and active research topic in commutative algebra and algebraic geometry. While local cohomology modules contain many useful and subtle information, one fundamental obstacle in understanding them is that they are often not finitely generated. Let I be a graded ideal in a polynomial ring over a field. In this talk I will describe a recent work with Jonathan Montano on estimating the Hilbert functions of the local cohomologies of powers of I. One particularly interesting and crucial case is when I is a monomial ideal, where it turns out that the counting function for the graded pieces can be described using Presburger arithmetic.