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

Synopsis

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).

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

See also

Ways to use multLowerBound :