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veronese -- Veronese embedding

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 : veronese(1,4)

                                    4   3     2 2     3   4
o1 = map (QQ[t ..t ], QQ[x ..x ], {t , t t , t t , t t , t })
              0   1       0   4     0   0 1   0 1   0 1   1

o1 : RingMap QQ[t ..t ] <-- QQ[x ..x ]
                 0   1          0   4
 
i2 : veronese(1,4,Variable=>y)

                                    4   3     2 2     3   4
o2 = map (QQ[y ..y ], QQ[y ..y ], {y , y y , y y , y y , y })
              0   1       0   4     0   0 1   0 1   0 1   1

o2 : RingMap QQ[y ..y ] <-- QQ[y ..y ]
                 0   1          0   4
 
i3 : veronese(1,4,Variable=>(u,z))

                                    4   3     2 2     3   4
o3 = map (QQ[u ..u ], QQ[z ..z ], {u , u u , u u , u u , u })
              0   1       0   4     0   0 1   0 1   0 1   1

o3 : RingMap QQ[u ..u ] <-- QQ[z ..z ]
                 0   1          0   4
 
i4 : veronese(2,2,ZZ/101)

           ZZ           ZZ            2               2         2
o4 = map (---[t ..t ], ---[x ..x ], {t , t t , t t , t , t t , t })
          101  0   2   101  0   5     0   0 1   0 2   1   1 2   2

              ZZ              ZZ
o4 : RingMap ---[t ..t ] <-- ---[x ..x ]
             101  0   2      101  0   5

See also

Ways to use veronese:

  • veronese(ZZ,ZZ)
  • veronese(ZZ,ZZ,Ring)

For the programmer

The object veronese is a method function with options.

The source of this document is in Resultants.m2:1000:0.