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reductionNumber -- reduction number of an ideal

Synopsis

Description

reductionNumber takes an ideal I that is homogeneous or inhomogeneous (in the latter case the ideal is to be regarded as an ideal in the localization of the polynomial ring at the origin.). It returns the integer k such that for a generic minimal reduction J of I, JIk = Ik+1.

The routine is probabilistic, since it depends on the routine minimalReduction.

See the book Huneke, Craig; Swanson, Irena: Integral closure of ideals, rings, and modules. London Mathematical Society Lecture Note Series, 336. Cambridge University Press, Cambridge, 2006. for further information.
i1 : kk = ZZ/101;
i2 : S = kk[a..c];
i3 : m = ideal vars S;

o3 : Ideal of S
i4 : i = (ideal"a,b")*m+ideal"c3"

             2                  2        3
o4 = ideal (a , a*b, a*c, a*b, b , b*c, c )

o4 : Ideal of S
i5 : analyticSpread i

o5 = 3
i6 : minimalReduction i
Warning: minimal reduction is not necessarily homogeneous

               3      2              2                    3      2        
o6 = ideal (50c  + 42a  - 41a*b - 15b  + 39a*c - 22b*c, 2c  + 45a  + a*b +
     ------------------------------------------------------------------------
        2                       3     2              2
     19b  - 39a*c - 38b*c, - 32c  - 4a  - 42a*b - 32b  - 16a*c + 31b*c)

o6 : Ideal of S
i7 : reductionNumber i
Warning: minimal reduction is not necessarily homogeneous

o7 = 1

See also

Ways to use reductionNumber :