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multBounds -- determine whether the multiplicity of an ideal satisfies the upper and lower bounds from the conjectures of Herzog-Huneke-Srinivasan

Synopsis

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.

S=ZZ/32003[a..c];
multBounds ideal(a^4,b^4,c^4)
multBounds ideal(a^3,b^4,c^5,a*b^3,b*c^2,a^2*c^3)

See also

Ways to use multBounds :