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RandomPoints :: Replacement

Replacement -- When changing coordinates, whether to replace variables by general degre 1 forms, binomials, etc.

Synopsis

Description

When calling various functions, setting Replacement => Full will mean that coordinates are changed to a general degree 1 form. If Replacement => Binomial, the coordinates are only changed to bionomials, which can be much faster for certain applications. Other options include Replacement => Monomial and Replacement => Trinomial.

i1 : R = ZZ/101[a,b,c,d,e];
 
i2 : randomCoordinateChange(R, Replacement=>Monomial)

              ZZ
o2 = map (R, ---[a..e], {a, d, e, b, c})
             101

                     ZZ
o2 : RingMap R <--- ---[a..e]
                    101
 
i3 : randomCoordinateChange(R, Replacement=>Binomial)

              ZZ
o3 = map (R, ---[a..e], {- 16a + d, -12c, - 24d + e, a + 19d, b + 21c})
             101

                     ZZ
o3 : RingMap R <--- ---[a..e]
                    101
 
i4 : randomCoordinateChange(R, Replacement=>Trinomial)

              ZZ
o4 = map (R, ---[a..e], {22b + 45d + e, 47b + 40d, 11b + c + 36d, - 11a + d + 43e, a - 16b - 48d})
             101

                     ZZ
o4 : RingMap R <--- ---[a..e]
                    101
 
i5 : randomCoordinateChange(R, Replacement=>Full)

              ZZ
o5 = map (R, ---[a..e], {- 10a + 30b - 18c + 39d + 27e, 24a - 30b - 48c - 15d + 39e, 29a - 47b + 15c - 37d - 13e, - 22a + 32b - 9c - 32d - 20e, 33b - 49c - 33d - 19e})
             101

                     ZZ
o5 : RingMap R <--- ---[a..e]
                    101

If Homogeneous => false in these cases, then there will be constant terms, and we view $mx + b$ as a monomial.

i6 : S = ZZ/103[x,y,z,u,v];
 
i7 : randomCoordinateChange(S, Replacement => Monomial, Homogeneous => false)

              ZZ
o7 = map (S, ---[x..z, u..v], {z + 4, y + 9, x + 13, v - 28, u - 41})
             103

                     ZZ
o7 : RingMap S <--- ---[x..z, u..v]
                    103
 
i8 : randomCoordinateChange(S, Replacement => Binomial, Homogeneous => false)

              ZZ
o8 = map (S, ---[x..z, u..v], {- 32y + 3, - 32y + u - 24, z - 30u + 16, x - 43v - 4, - 37x + v - 11})
             103

                     ZZ
o8 : RingMap S <--- ---[x..z, u..v]
                    103
 
i9 : randomCoordinateChange(S, Replacement => Trinomial, Homogeneous => false)

              ZZ
o9 = map (S, ---[x..z, u..v], {- 14u + 30v + 7, - 20y + 41z + 28, - 19x + 42y + 49, 30x + 48v - 50, 8x + y - 31z - 43})
             103

                     ZZ
o9 : RingMap S <--- ---[x..z, u..v]
                    103
 
i10 : randomCoordinateChange(S, Replacement => Full, Homogeneous => false)

               ZZ
o10 = map (S, ---[x..z, u..v], {6x - 11y - 35z + 28u - 31v - 30, 30x + 4y + 22z + 5u - 22v + 44, 19x - 49z - 30u + 47v + 20, 5x - 39y - 35z - 30u + 42v + 26, 23x - 39y - 25z + 44u - 41v - 23})
              103

                      ZZ
o10 : RingMap S <--- ---[x..z, u..v]
                     103

See also

Functions with optional argument named Replacement :

For the programmer

The object Replacement is a symbol.