calcResultant -- Calculate resultant of sparse system.

Usage:

Res = calcResultant( f, Lift, shiftP )
Inputs:
- f, a list, List of sparse polynomials (specified in the algebraic torus)
- Lift, a list, additional coordinate to lift the vertices of the system's Newton polytopes
- shiftP, a list, QQ-coordinates of Canny Emiris vector to shift Minkowski sums of Newton polytopes
Outputs:
- Res, a ring element, resultant of the sparse system contained in the (multi-homogeneous) ring of coefficients

Description

Once we have chosen lifting vectors which produce a tight coherent subdivision of the system's Newton polytopes and specified a Canny Emiris vector, the resultant of the sparse system can be calculated as determinant of a complex via Cayley formula resultantComplex and CayleyFormula. As the number of terms in resultants can grow rapidly, this method is only useful for smaller systems. Except that the Minkowski sum of the Newton polytopes must be full dimensional, there are no restrictions on the family of support sets. Hence resultants may have multiplicities, they agree with the definition of D'Andrea and Sombra. Further, 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 sets.

i1 : supp = { {{0,0},{2,2},{1,3}}, {{0,0},{2,0},{1,2}}, {{3,0},{1,1}} };

i2 : A = QQ[c_1..c_8];

i3 : R = A[t_1..t_2];

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

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

o4 : List

i5 : Res = calcResultant( f, {{-1, 0, 0},{-7, -13, 0},{0, 0}}, {0,1/3} )

      5 7 7       4 5     6       2   2 4 3 6      5 7 6      2 3 6   5 2  
o5 = c c c  - 7c c c c c c c  + 5c c c c c c c  + c c c c  + c c c c c c  -
      3 4 7     1 3 4 5 6 7 8     1 2 3 4 6 7 8    1 6 7 8    2 3 4 5 7 8  
     ------------------------------------------------------------------------
         3   5 2 5 2      2 3 3 2 2 5 2      3     2   4 5 2    5 6   4 3  
     5c c c c c c c  + 14c c c c c c c  - 13c c c c c c c c  + c c c c c  +
       1 2 3 4 6 7 8      1 3 4 5 6 7 8      1 2 3 4 5 6 7 8    2 4 6 7 8  
     ------------------------------------------------------------------------
         2 2 4 2   4 3     2 3 3   3 4 3     3 2   3 3 4 3     4   2 5 4 3  
     2c c c c c c c c  + 6c c c c c c c  - 7c c c c c c c  + 3c c c c c c  -
       1 2 3 4 5 6 7 8     1 2 4 5 6 7 8     1 3 4 5 6 7 8     1 2 5 6 7 8  
     ------------------------------------------------------------------------
           3 3 4 3 4    2 2   2 3 2 3 4       4 3 3   2 5    2   2   5   2 5
     2c c c c c c c  - c c c c c c c c  - 2c c c c c c c  - c c c c c c c c 
       1 2 3 4 5 7 8    1 2 3 4 5 6 7 8     1 2 4 5 6 7 8    1 2 3 4 5 6 7 8
     ------------------------------------------------------------------------
         3 2 4 3 2 5    2 3 7   6    2 3 6   7
     + 3c c c c c c  + c c c c c  + c c c c c
         1 2 5 6 7 8    1 3 5 7 8    1 2 5 6 8

o5 : A

Ways to use calcResultant:

calcResultant(List,List,List)

For the programmer

The object calcResultant is a method function.

The source of this document is in ResultantComplexes.m2:440:0.