multidegree -- multidegree

Synopsis

• Usage:

  multidegree M

• Inputs:
  • M, a module, an ideal, or a ring
• Outputs:
  • the multidegree of M. If M is an ideal, the corresponding quotient ring is used.

Description

The multidegree is defined on page 165 of Combinatorial Commutative Algebra, by Miller and Sturmfels. It is an element of the degrees ring of M. Our implementation agrees with their definition provided the heft vector of the ring has every entry equal to 1. See also Gröbner geometry of Schubert polynomials, by Allen Knutson and Ezra Miller.

i1 : S = QQ[a..d, Degrees => {{2,-1},{1,0},{0,1},{-1,2}}];

i2 : heft S

o2 = {1, 1}

o2 : List

i3 : multidegree ideal (b^2,b*c,c^2)

o3 = 3T T
       0 1

o3 : ZZ[T ..T ]
         0   1

i4 : multidegree ideal a

o4 = 2T  - T
       0    1

o4 : ZZ[T ..T ]
         0   1

i5 : multidegree ideal (a^2,a*b,b^2)

       2
o5 = 6T  - 3T T
       0     0 1

o5 : ZZ[T ..T ]
         0   1

i6 : describe ring oo

o6 = ZZ[T ..T , Degrees => {2:1}, MonomialOrder => {MonomialSize => 32}, Inverses => true, Global => false]
         0   1                                     {Weights => {2:-1} }
                                                   {GroupLex => 2     }
                                                   {Position => Up    }

Caveat

See also

• heft vectors
• degreesRing -- the ring or monoid of degrees