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

Synopsis

• Usage:
Replacement => Full
Replacement => Binomial

Description

When calling randomCoordinateChange, or functions that call it, setting Replacement => Full will mean that coordinates are changed to a general degree 1 form. If Replacement => Binomial, the coordiates are only changed to bionomials, which can be much faster for certain applications.

 i1 : R = ZZ/11[a,b,c]; i2 : randomCoordinateChange(R, Replacement=>Full) ZZ o2 = map(R,--[a..c],{3a - 4b - 3c, - 3a - b + 3c, a + 3b - 4c}) 11 ZZ o2 : RingMap R <--- --[a..c] 11 i3 : randomCoordinateChange(R, Replacement=>Binomial) ZZ o3 = map(R,--[a..c],{-5c, -3a, b + 3c}) 11 ZZ o3 : RingMap R <--- --[a..c] 11

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

 i4 : S = ZZ/11[x,y]; i5 : randomCoordinateChange(S, Replacement => Full, Homogeneous => false) ZZ o5 = map(S,--[x..y],{- 5x - 2y - 1, - 4x - 5y + 3}) 11 ZZ o5 : RingMap S <--- --[x..y] 11 i6 : randomCoordinateChange(S, Replacement => Binomial, Homogeneous => false) ZZ o6 = map(S,--[x..y],{- 5x + y + 2, x - 5y + 2}) 11 ZZ o6 : RingMap S <--- --[x..y] 11

Functions with optional argument named Replacement :

• "extendIdealByNonZeroMinor(...,Replacement=>...)"
• "findANonZeroMinor(...,Replacement=>...)"
• "genericProjection(...,Replacement=>...)"
• "projectionToHypersurface(...,Replacement=>...)"
• "randomCoordinateChange(...,Replacement=>...)"
• "randomPoints(...,Replacement=>...)"

For the programmer

The object Replacement is .