# mfBound -- determines how high a syzygy to take for "matrixFactorization"

## Synopsis

• Usage:
p = mfBound M
• Inputs:
• M, , over a complete intersection
• Outputs:

## Description

If p = mfBound M, then the p-th syzygy of M, which is computed by highSyzygy(M), should (this is a conjecture) be a "high Syzygy" in the sense required for matrixFactorization. In examples, the estimate seems sharp (except when M is already a high syzygy).

The actual formula used is:

mfBound M = max(2*r_{even}, 1+2*r_{odd})

where r_{even} = regularity evenExtModule M and r_{odd} = regularity oddExtModule M. Here evenExtModule M is the even degree part of Ext(M, (residue class field)).