genericLaurentPolynomials -- generic (Laurent) polynomials

Synopsis

• Usage:

  genericLaurentPolynomials A

• Inputs:
  • A, a sequence, or a list of $n+1$ matrices $A_0,\ldots,A_n$ over $\mathbb{Z}$ and with $n$ rows to represent the exponent vectors of (Laurent) polynomials $f_0,\ldots,f_n$ in $n$ variables. For the dense case, one can pass just the sequence of degrees of $f_0,\ldots,f_n$.
• Optional inputs:
  • CoefficientRing => ..., default value ZZ
• Outputs:
  • the generic (Laurent) polynomials $f_0,\ldots,f_n$ in the ring $\mathbb{Z}[a_0,a_1,\ldots,b_0,b_1,\ldots][x_1,\ldots,x_n]$, involving only the monomials from $A$. (Note that, if all the exponents are nonnegative, then the ambient polynomial ring is taken without inverses of variables, so that $f_0,\ldots,f_n$ are ordinary polynomials.)

Description

This method helps to construct special types of sparse resultants, see for instance denseResultant.

i1 : M = (matrix{{2,3,4,5},{0,2,1,0}},matrix{{1,-1,0,2,3},{-2,0,-7,-1,0}},matrix{{-1,0,6},{-2,1,3}})

o1 = (| 2 3 4 5 |, | 1  -1 0  2  3 |, | -1 0 6 |)
      | 0 2 1 0 |  | -2 0  -7 -1 0 |  | -2 1 3 |

o1 : Sequence

i2 : genericLaurentPolynomials M

         5      4        3 2      2     3      2 -1        -2      -7  
o2 = (a x  + a x x  + a x x  + a x , b x  + b x x   + b x x   + b x   +
       3 1    2 1 2    1 1 2    0 1   4 1    3 1 2     2 1 2     1 2   
     ------------------------------------------------------------------------
        -1     6 3             -1 -2
     b x  , c x x  + c x  + c x  x  )
      0 1    2 1 2    1 2    0 1  2

o2 : Sequence

i3 : genericLaurentPolynomials (2,3,1)

         2               2                        3      2          2      3
o3 = (a x  + a x x  + a x  + a x  + a x  + a , b x  + b x x  + b x x  + b x 
       5 1    4 1 2    2 2    3 1    1 2    0   9 1    8 1 2    6 1 2    3 2
     ------------------------------------------------------------------------
          2               2
     + b x  + b x x  + b x  + b x  + b x  + b , c x  + c x  + c )
        7 1    5 1 2    2 2    4 1    1 2    0   2 1    1 2    0

o3 : Sequence

See also

• sparseResultant -- sparse resultant (A-resultant)
• exponentsMatrix -- exponents in one or more polynomials