genericPolynomials -- generic homogeneous polynomials

Synopsis

Usage:

genericPolynomials(d,K)

genericPolynomials d
Inputs:
- d, a list, n+1 integers d_0,...,d_n
- K, a ring, optional, with default value QQ
Outputs:
- a list, n+1 generic homogeneous polynomials of degrees d_0,...,d_n in the ring K[a_0,a_1,...,b_0,b_1,...][x_0,...,x_n]

Description

This is an auxiliary method to build tests and examples. For instance, the two following codes have to produce the same polynomial up to a renaming of variables: 1) resultant genericPolynomials((n+1):d,K) and 2) fromPluckerToStiefel dualize chowForm veronese(n,d,K).

i1 : genericPolynomials {1,2,3}

                             2               2                        2     3
o1 = {a x  + a x  + a x , b x  + b x x  + b x  + b x x  + b x x  + b x , c x 
       0 0    1 1    2 2   0 0    1 0 1    3 1    2 0 2    4 1 2    5 2   0 0
     ------------------------------------------------------------------------
          2          2      3      2                   2          2        2
     + c x x  + c x x  + c x  + c x x  + c x x x  + c x x  + c x x  + c x x 
        1 0 1    3 0 1    6 1    2 0 2    4 0 1 2    7 1 2    5 0 2    8 1 2
     ------------------------------------------------------------------------
          3
     + c x }
        9 2

o1 : List

i2 : first genericPolynomials({4,2,3},ZZ/101)

        4      3        2 2        3       4      3        2            2    
o2 = a x  + a x x  + a x x  + a x x  + a  x  + a x x  + a x x x  + a x x x  +
      0 0    1 0 1    3 0 1    6 0 1    10 1    2 0 2    4 0 1 2    7 0 1 2  
     ------------------------------------------------------------------------
         3        2 2          2       2 2        3         3       4
     a  x x  + a x x  + a x x x  + a  x x  + a x x  + a  x x  + a  x
      11 1 2    5 0 2    8 0 1 2    12 1 2    9 0 2    13 1 2    14 2

      ZZ
o2 : ---[a ..a  , b ..b , c ..c ][x ..x ]
     101  0   14   0   5   0   9   0   2

i3 : first genericPolynomials({4,-1,-1},ZZ/101)

        4      3        2 2        3       4      3        2            2    
o3 = a x  + a x x  + a x x  + a x x  + a  x  + a x x  + a x x x  + a x x x  +
      0 0    1 0 1    3 0 1    6 0 1    10 1    2 0 2    4 0 1 2    7 0 1 2  
     ------------------------------------------------------------------------
         3        2 2          2       2 2        3         3       4
     a  x x  + a x x  + a x x x  + a  x x  + a x x  + a  x x  + a  x
      11 1 2    5 0 2    8 0 1 2    12 1 2    9 0 2    13 1 2    14 2

      ZZ
o3 : ---[a ..a  ][x ..x ]
     101  0   14   0   2

Ways to use genericPolynomials :

genericPolynomials(List)
genericPolynomials(VisibleList,Ring)

genericPolynomials -- generic homogeneous polynomials

Synopsis

Description

See also

Ways to use genericPolynomials :

For the programmer