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multUpperBound -- determine whether the multiplicity of an ideal satisfies the upper bound conjectured by Herzog-Huneke-Srinivasan

Synopsis

Description

Let I be a homogeneous ideal of codimension c in a polynomial ring R. Huneke and Srinivasan (and later Herzog and Srinivasan in the non-Cohen-Macaulay case) conjectured that

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

where 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. multUpperBound tests this inequality for the given ideal, returning true if the inequality holds and false otherwise, and it prints the upper bound and the multiplicity (listed as the degree).

i1 : R=ZZ/32003[a..c];
i2 : multUpperBound ideal(a^4,b^4,c^4)

degree = 64  upper bound = 64

o2 = true
i3 : multUpperBound ideal(a^3,b^5,c^6,a^2*b,a*b*c)

degree = 46  upper bound = 132

o3 = true

See also

Ways to use multUpperBound :