random(ZZ,Ideal) -- get a random homogeneous element from a graded ideal

Synopsis

• Function: random

• Usage:

  random(d, I)

  random(L, I)

• Inputs:
  • d, an integer, an integer or a list, the degree, multi-degree, or a list of degrees;
  • I, an ideal, in a graded ring;
• Optional inputs:
  • Density => ..., default value 1, get a random map of module
  • Height => ..., default value 10, get a random homogeneous element from a graded ring
  • MaximalRank => ..., default value false, get a random map of module
  • UpperTriangular => ..., default value false, get a random map of module
• Outputs:
  • a ring element, a ring element or a list, homogeneous element(s) with prescribed degrees;

Description

i1 : S = ZZ/101[x, y]

o1 = S

o1 : PolynomialRing

i2 : I = ideal"x2, y2"

             2   2
o2 = ideal (x , y )

o2 : Ideal of S

i3 : random(2, I)

        2      2
o3 = 24x  - 36y

o3 : S

i4 : random({2}, I)

          2      2
o4 = - 30x  - 29y

o4 : S

i5 : random({2, 3}, I)

         2      2       3     2         2      3
o5 = {19x  + 19y , - 10x  - 8x y - 29x*y  - 22y }

o5 : List

i6 : random({{2}, {3}}, I)

           2      2       3      2         2      3
o6 = {- 29x  - 24y , - 38x  + 39x y - 16x*y  + 21y }

o6 : List

i7 : R = ZZ/101[x, y, z, Degrees => {{1,0}, {-1,1}, {0,1}}]

o7 = R

o7 : PolynomialRing

i8 : J = ideal"x2, y2, z2"

             2   2   2
o8 = ideal (x , y , z )

o8 : Ideal of R

i9 : random({2, 2}, J)

         4 2      3         2 2
o9 = - 5x y  + 19x y*z + 36x z

o9 : R

i10 : random({{2, 2}}, J)

            4 2      3         2 2
o10 = {- 41x y  - 43x y*z + 39x z }

o10 : List

i11 : random(toList(3:{1, 1}), J)

          2     2      2
o11 = {38x y, 2x y, 16x y}

o11 : List

Caveat

See also

• random(ZZ,Ring) -- get a random homogeneous element from a graded ring
• random(List,Ring) -- get a random homogeneous element from a graded ring