multLowerBound -- determine whether an ideal satisfies the lower bound of the multiplicity conjecture

Synopsis

Usage:

B=multLowerBound I
Inputs:
- I, an ideal, a homogeneous ideal in a polynomial ring R
Outputs:
- B, a Boolean value, true if I satisfies the lower bound and false otherwise

Description

Let I be a homogeneous ideal of codimension c in a polynomial ring R such that R/I is Cohen-Macaulay. Huneke and Srinivasan conjectured that

m_1 ... m_c / c! <= e(R/I),

where m_i is the minimum shift in the minimal graded free resolution of R/I at step i, and e(R/I) is the multiplicity of R/I. multLowerBound tests this inequality for the given ideal, returning true if the inequality holds and false otherwise, and it prints the lower bound and the multiplicity (listed as the degree).

This conjecture was proven in 2008 work of Eisenbud-Schreyer and Boij-Soderberg.

i1 : R=ZZ/32003[a..c];

i2 : multLowerBound ideal(a^4,b^4,c^4)

lower bound = 64 degree = 64

o2 = true

i3 : multLowerBound ideal(a^3,b^5,c^6,a^2*b,a*b*c)

lower bound = 16 degree = 46

o3 = true

Caveat

Note that multLowerBound makes no attempt to check to see whether R/I is Cohen-Macaulay.

Ways to use multLowerBound :

multLowerBound(Ideal)

For the programmer

The object multLowerBound is a method function.