genericMultihomogeneousPolynomial -- generic multi-homogeneous polynomial

Usage:

genericMultihomogeneousPolynomial((k_1,...,k_n),(d_1,...,d_n))
Inputs:
- (k_1,...,k_n), a sequence of positive integers to indicate $n$ sets of variables $X_1,\ldots,X_n$ with $\#(X_i) = k_i$.
- (d_1,...,d_n), a sequence of nonnegative integers.
Optional inputs:
- CoefficientRing => ..., default value ZZ
- Variable => ..., default value "a"
Outputs:
- a ring element, the generic multi-homogeneous polynomial of multi-degree $(d_1,\ldots,d_n)$ in the above sets of variables.

Description

This method helps to construct special types of sparse discriminants. For instance, the hyperdeterminant of a generic $(k_1\times\cdots\times k_n)$-matrix can be obtained with the code: sparseDiscriminant genericMultihomogeneousPolynomial((k_1,...,k_n),(1,...,1)).

i1 : genericMultihomogeneousPolynomial((2,2,3),(1,1,1))

o1 = a     x y z  + a     x y z  + a     x y z  + a     x y z  + a     x y z 
      0,0,0 0 0 0    0,0,1 0 0 1    0,0,2 0 0 2    0,1,0 0 1 0    0,1,1 0 1 1
     ------------------------------------------------------------------------
     + a     x y z  + a     x y z  + a     x y z  + a     x y z  +
        0,1,2 0 1 2    1,0,0 1 0 0    1,0,1 1 0 1    1,0,2 1 0 2  
     ------------------------------------------------------------------------
     a     x y z  + a     x y z  + a     x y z
      1,1,0 1 1 0    1,1,1 1 1 1    1,1,2 1 1 2

o1 : ZZ[a     ..a     ][x ..y , z ..z ]
         0,0,0   1,1,2   0   1   0   2

i2 : genericMultihomogeneousPolynomial((2,3),(3,1))

          3          3          3          2            2            2      
o2 = a   x y  + a   x y  + a   x y  + a   x x y  + a   x x y  + a   x x y  +
      0,0 0 0    0,1 0 1    0,2 0 2    1,0 0 1 0    1,1 0 1 1    1,2 0 1 2  
     ------------------------------------------------------------------------
            2            2            2          3          3          3
     a   x x y  + a   x x y  + a   x x y  + a   x y  + a   x y  + a   x y
      2,0 0 1 0    2,1 0 1 1    2,2 0 1 2    3,0 1 0    3,1 1 1    3,2 1 2

o2 : ZZ[a   ..a   ][x ..x , y ..y ]
         0,0   3,2   0   1   0   2

i3 : genericMultihomogeneousPolynomial((2,2),(1,1),CoefficientRing=>ZZ/33331)

o3 = a   x y  + a   x y  + a   x y  + a   x y
      0,0 0 0    0,1 0 1    1,0 1 0    1,1 1 1

       ZZ
o3 : -----[a   ..a   ][x ..y ]
     33331  0,0   1,1   0   1

Ways to use genericMultihomogeneousPolynomial:

genericMultihomogeneousPolynomial(VisibleList,VisibleList)

For the programmer

The object genericMultihomogeneousPolynomial is a method function with options.

The source of this document is in SparseResultants.m2:1239:0.

genericMultihomogeneousPolynomial -- generic multi-homogeneous polynomial

Description

See also

Ways to use genericMultihomogeneousPolynomial:

For the programmer