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FiniteFittingIdeals :: gotzmannTest

gotzmannTest -- Checks if a set of monomials is a Gotzmann set



For us, a Gotzmann set will be a set L of monomials of degree d in the variables x_0,\dots,x_r with the property that if m\in L, then x_0 divides m and if x_i divides m, then (x_0m)/x_i\in L. The function gotzmannTest checks if a set of monomials fulfills this property.

i1 : S=ZZ[x,y,z];
i2 : L={x^3,x^2*y,x^2*z,x*y*z}

       3   2    2
o2 = {x , x y, x z, x*y*z}

o2 : List
i3 : gotzmannTest(L,x)

o3 = true

A non example of a Gotzmann set is L_2=\{x^3,x^2y,xz^2\}.

i4 : L2={x^3,x^2*y,x*z^2}

       3   2      2
o4 = {x , x y, x*z }

o4 : List
i5 : gotzmannTest(L2,x)

o5 = false

L_2 is not a Gotzmann set since it does not contain x^2z.

When we consider a free S-module S^p with basis e_1,\dots,e_p, then we generalize our notion of Gotzmann set for x so that a set L is a Gotzmann set if it is a union of Gotzmann sets for x for e_1,\dots,e_p.

As an example in S^2=\mathbb{Z}[x,y,z]^2 we have a Gotzmann set L=\{x^2e_1,xye_1,x^2e_2\}\ as it is a Gotzmann set in each coordinate. We can test this be gotzmannTest(S^p,d,I), where d is the degree of the monomials, and I is the index of the monomials of L listed in the lexicographical order x<y<z<e_1<e_2. In our case we have d=2 and I=\{0,1,6\}\ since:


i6 : gotzmannTest(S^2,2,{0,1,6})

o6 = true

See also

Ways to use gotzmannTest :

For the programmer

The object gotzmannTest is a method function.