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



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


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

See also

Ways to use multBounds :

For the programmer

The object multBounds is a method function.