PolynomialMapImage - Maple Help

RegularChains[ConstructibleSetTools]

 PolynomialMapImage
 compute the image of a variety under a polynomial map

 Calling Sequence PolynomialMapImage(F, PM, R, S) PolynomialMapImage(F, H, PM, R, S) PolynomialMapImage(CS, PM, R, S)

Parameters

 F - list of polynomials in R PM - list of polynomials in R R - polynomial ring (source) S - polynomial ring (target) H - list of polynomials in R CS - constructible set

Description

 • The command PolynomialMapImage(F, PM, R, S) returns a constructible set cs which is the image of the variety $V\left(F\right)$ under the polynomial map PM.
 • The command PolynomialMapImage(F, H, PM, R, S) returns a constructible set cs which is the image of the difference of the variety $V\left(F\right)$ by the variety $V\left(H\right)$ under the polynomial map PM.
 • The command PolynomialMapImage(CS, PM, R, S) returns a constructible set cs which is the image of the constructible set CS under the polynomial map PM.
 • Both rings R and S should be over the same base field.
 • The variable sets of R and S should be disjoint.
 • The number of polynomials in PM is equal to the number of variables of ring S.
 • This command is part of the RegularChains[ConstructibleSetTools] package, so it can be used in the form PolynomialMapImage(..) only after executing the command with(RegularChains[ConstructibleSetTools]).  However, it can always be accessed through the long form of the command by using RegularChains[ConstructibleSetTools][PolynomialMapImage](..).

Examples

 > $\mathrm{with}\left(\mathrm{RegularChains}\right):$
 > $\mathrm{with}\left(\mathrm{ConstructibleSetTools}\right):$

The following example is related to the Whitney umbrella.

 > $R≔\mathrm{PolynomialRing}\left(\left[u,v\right]\right)$
 ${R}{≔}{\mathrm{polynomial_ring}}$ (1)
 > $S≔\mathrm{PolynomialRing}\left(\left[x,y,z\right]\right)$
 ${S}{≔}{\mathrm{polynomial_ring}}$ (2)
 > $\mathrm{PM}≔\left[uv,u,{v}^{2}\right]$
 ${\mathrm{PM}}{≔}\left[{u}{}{v}{,}{u}{,}{{v}}^{{2}}\right]$ (3)
 > $\mathrm{cs}≔\mathrm{PolynomialMapImage}\left(\left[\right],\mathrm{PM},R,S\right)$
 ${\mathrm{cs}}{≔}{\mathrm{constructible_set}}$ (4)
 > $\mathrm{cs}≔\mathrm{MakePairwiseDisjoint}\left(\mathrm{cs},S\right)$
 ${\mathrm{cs}}{≔}{\mathrm{constructible_set}}$ (5)
 > $\mathrm{Info}\left(\mathrm{cs},S\right)$
 $\left[\left[{x}{,}{y}\right]{,}\left[{1}\right]\right]{,}\left[\left[{{x}}^{{2}}{-}{{y}}^{{2}}{}{z}\right]{,}\left[{y}\right]\right]$ (6)