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lexIdeal -- produce an Artinian lexicographic ideal

Synopsis

Description

When R is a polynomial ring, if hilb is an O-sequence (that is, it satisfies Macaulay's Theorem), such an L always exists. When Q is a quotient of a polynomial ring, there may be no lexicographic ideal with a particular Hilbert function even if it is an O-sequence. lexIdeal returns null if no lexicographic ideal L corresponding to the Hilbert function hilb exists in R or Q.

We hope eventually to implement a version of lexIdeal for nonArtinian ideals, taking a Hilbert series as the input.

i1 : R=ZZ/32003[a..c];
i2 : lexIdeal(R,{1,3,4,3,1})

             2          2   3   2 2     3   5
o2 = ideal (a , a*b, a*c , b , b c , b*c , c )

o2 : Ideal of R
i3 : lexIdeal(R,{1,3,7}) --not an O-sequence, so no lex ideal exists
i4 : Q=R/ideal(a^3,b^3,a*c^2);
i5 : lexIdeal(Q,{1,3,6,4,2})

             2      2   2    2 2   5     4
o5 = ideal (a c, a*b , a b, b c , c , b*c )

o5 : Ideal of Q
i6 : lexIdeal(Q,{1,3,6,4,4}) --value of 4 in degree 4 is too high in this ring

See also

Ways to use lexIdeal :