Project: Integral Closure

• Potential advisor/consultant(s): David Eisenbud and Mike Stillman
• Goal: Fast integral closure with good interface and output
• Current status: Many subprograms and packages written; needs consolidation. Collaborators welcome.
• Macaulay2 skill level: Intermediate
• Mathematical experience: Grad commutative algebra, algebraic geometry desirable

Project Description

Fractional ideals and Noether normalization -- smooth out interface

Puiseux series and integral bases for plane curves (and a version for finite characteristic with Frobenius)

Fast integral closure of plane curves will be a milestone

Need easy ways to adjoin roots in fields

Work over finite fields, lift to char 0

Work over function fields (Hensel lifting?)

Early milestone: unify pushfwd (in the PushForward package) with the internal pushForward command

Examples

Existing packages

##Published IntegralClosure

ReesAlgebra

Riemann-Roch spaces

NoetherNormalization ##On one of Mike's branches 2018-Mike-packages in mikestillman IntegralBases AdjoinRoots Puiseux FractionalIdeals LocalBasis? (with Mark van Hoeij) integral-closure -- contains many odds and ends

##On eisenbud/M2 there is a branch david-mike that has NoetherNormalForm FractionalIdeals

References (that describe the math and/or algorithms)