randomLift -- Get random lift-vectors to produce coherent subdivisions.

Usage:

Lift = randomLift( f, m )
Inputs:
- f, a list, of polynomials
- m, an integer, range of random numbers assigned to vertices of Newton polytopes
Outputs:
- Lift, a list, values of lifting function at the vertices of Newton polytopes

Description

Randomly selects additional coordinates for the list of points whose sub-division is to be determined. Tight coherent mixed sub-divisions (TCMD) determine the admissible sub-modules of the resultant complex required for the Cayley formula: CayleyFormula and mixedSubdivision. To reduce the number of cells in the sub-division, we specify the lift function only at the vertices of the Newton polytopes and not on the entire support set. Randomly chosen liftings almost always result in tight mixed sub-divisions.

i1 : supp = {{{2,1,0},{0,2,4},{4,2,0}},{{1,0,0},{2,2,0}},{{0,1,2},{0,0,0},{0,2,4}},{{0,1,0},{0,0,2}}};

i2 : A = QQ[c_1..c_(#flatten supp)];

i3 : R = A[t_1..t_(#(supp#0#0))];

i4 : f = polynomials(gens A, gens R, supp)

         4 2      2 4      2       2 2            2 4        2           2
o4 = {c t t  + c t t  + c t t , c t t  + c t , c t t  + c t t  + c , c  t  +
       3 1 2    2 2 3    1 1 2   5 1 2    4 1   8 2 3    6 2 3    7   10 3  
     ------------------------------------------------------------------------
     c t }
      9 2

o4 : List

i5 : Lift = randomLift(f, 80)

o5 = {{-56, -15, -9}, {-8, -61}, {-61, -8}, {-1, -8}}

o5 : List

Ways to use randomLift:

randomLift(List,ZZ) (missing documentation)

For the programmer

The object randomLift is a method function.

The source of this document is in ResultantComplexes.m2:549:0.