findMonomialSubalgebra -- Find monomial subalgebra corresponding to the convex hull.

Synopsis

Usage:

findMonomialSubalgebra M
Inputs:
- R, a polynomial ring, with B = degrees R and K = coefficientRing R, or
- M, a MonomialAlgebra,
Outputs:
- a polynomial ring,
- a MonomialAlgebra,

Description

Denote by C(B) the cone in \mathbb{R}^d spanned by B. This function computes on each ray of C(B) one element of B which has minimal coordinate sum, and returns the multigraded polynomial ring with the corresponding variables.

If a monomial algebra is specified the function returns a monomial algebra.

i1 : a=3

o1 = 3

i2 : B={{a, 0}, {0, a}, {1, a-1}, {a-1, 1}}

o2 = {{3, 0}, {0, 3}, {1, 2}, {2, 1}}

o2 : List

i3 : R=QQ[x_0..x_3, Degrees=> B]

o3 = R

o3 : PolynomialRing

i4 : findMonomialSubalgebra R

o4 = QQ[x ..x ]
         0   1

o4 : PolynomialRing

i5 : a=3

o5 = 3

i6 : B={{a, 0}, {0, a}, {1, a-1}, {a-1, 1}}

o6 = {{3, 0}, {0, 3}, {1, 2}, {2, 1}}

o6 : List

i7 : M=monomialAlgebra B

      ZZ
o7 = ---[x ..x ]
     101  0   3

o7 : MonomialAlgebra generated by {{3, 0}, {0, 3}, {1, 2}, {2, 1}}

i8 : findMonomialSubalgebra M

      ZZ
o8 = ---[x ..x ]
     101  0   1

o8 : MonomialAlgebra generated by {{3, 0}, {0, 3}}

Ways to use findMonomialSubalgebra :

findMonomialSubalgebra(MonomialAlgebra)
findMonomialSubalgebra(PolynomialRing)

For the programmer

The object findMonomialSubalgebra is a method function.