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 GrÃ¶bner 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}, DegreeRank => 1, Inverses => true, Global => false] 0 1 {Weights => {2:-1} } {GroupLex => 2 } {Position => Up } |
The object multidegree is a method function.