Title:Computations with Frobenius powers
Abstract: Suppose F is a field of characteristic p and I and J are ideals in the polynomial ring F[x_1,...,x_n] in n variables over F. If f_1,...,f_r generate the ideal I and q=p^e is a power of p, then the e th Frobenius power of I is the ideal I^[q] generated by f_1^q,...,f_r^q. According to a conjecture of Katzmann, there is an integer B such that the x_n-degrees of the elements of the reduced Groebner basis for J+I^[q] with respect to the reverse lexicographic ordering are bounded above by Bq. The motivation for this conjecture comes from the study of tight closure of ideals in commutative algebra. In this talk I will discuss this connection, along with how a special case of this conjecture can be reduced to a problem about monoids, with positive results in some cases. This work is joint with Irena Swanson.