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
m1 ... mc / c! <= e(R/I) <= M1 ... Mc / c!,
where mi is the minimum shift in the minimal graded free resolution of R/I at step i, Mi 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.
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 |