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part(List,RingElement) -- sum of terms of a polynomial of a given degree(s)

Synopsis

Description

Synopsis

  • Usage:
    part(d,F)
    part_d F
  • Inputs:
    • d, of integers denoting a multidegree
    • F, an element in a polynomial ring
  • Outputs:
If the polynomial ring is singly graded (the default case), then d may be an integer denoting this degree.
i1 : R = QQ[a..d]

o1 = R

o1 : PolynomialRing
i2 : f = (a^2-b-1)*(c^3-b*d-2)

      2 3      3    2       3    2      2
o2 = a c  - b*c  - a b*d - c  + b d - 2a  + b*d + 2b + 2

o2 : R
i3 : part({3},f)

        3    2
o3 = - c  + b d

o3 : R
Here is an alternate syntax.
i4 : part_{3} f

        3    2
o4 = - c  + b d

o4 : R
In multigraded rings, degrees are lists of integers.
i5 : R = QQ[a..d,Degrees=>{{1,0},{0,1},{1,-1},{0,-1}}]

o5 = R

o5 : PolynomialRing
i6 : F = a^3 + (b*d+1)^2

      2 2    3
o6 = b d  + a  + 2b*d + 1

o6 : R
i7 : part_{0,0} F

      2 2
o7 = b d  + 2b*d + 1

o7 : R
Polynomial rings over other polynomial rings are multigraded, by default.
i8 : A = QQ[a,b,c]

o8 = A

o8 : PolynomialRing
i9 : B = A[x,y]

o9 = B

o9 : PolynomialRing
i10 : degree(a*x)

o10 = {1, 1}

o10 : List
i11 : part_{2,2} (a*x+b*y-1)^3

          2 2                2 2
o11 = - 3a x  - 6a*b*x*y - 3b y

o11 : B

See also