# multBounds -- determine whether an ideal satisfies the upper and lower bounds of the multiplicity conjecture

## Synopsis

• Usage:
B=multBounds I
• Inputs:
• I, an ideal, a homogeneous ideal in a polynomial ring
• Outputs:
• B, , true if both the upper and lower bounds hold 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. Herzog, Huneke, and Srinivasan conjectured that if R/I is Cohen-Macaulay, then

m_1 ... m_c / c! <= e(R/I) <= M_1 ... M_c / c!,

where m_i is the minimum shift in the minimal graded free resolution of R/I at step i, M_i is the maximum shift in the minimal graded free resolution of R/I at step i, and e(R/I) is the multiplicity of R/I. If R/I is not Cohen-Macaulay, the upper bound is still conjectured to hold. multBounds tests the inequalities for the given ideal, returning true if both inequalities hold and false otherwise. multBounds prints the bounds and the multiplicity (called the degree), and it calls multUpperBound and multLowerBound.

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

 i1 : S=ZZ/32003[a..c]; i2 : multBounds ideal(a^4,b^4,c^4) lower bound = 64 degree = 64 upper bound = 64 o2 = true i3 : multBounds ideal(a^3,b^4,c^5,a*b^3,b*c^2,a^2*c^3) 35 140 lower bound = -- degree = 27 upper bound = --- 2 3 o3 = true

## Caveat

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