RegularChains[ConstructibleSetTools] - Maple Programming Help

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RegularChains[ConstructibleSetTools]

 PolynomialMapPreimage
 compute the preimage of a variety under a polynomial map

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

Parameters

 F - list of polynomials of S 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 PolynomialMapPreimage(F, PM, R, S) returns a constructible set cs over R, which is the preimage of the variety V(F) under the polynomial map PM.
 • The command PolynomialMapPreimage(F, H, PM, R, S) returns a constructible set cs over R, which is the preimage of the difference of the variety V(F) by the variety $V\left(H\right)$ under the polynomial map PM.
 • The command PolynomialMapPreimage(CS, PM, R, S) returns a constructible set cs over R, which is the preimage of the constructible set CS under the polynomial map PM.
 • Both rings R and S should be over the same ground field.
 • The variable sets of R and S should be disjoint.
 • This command is part of the RegularChains[ConstructibleSetTools] package, so it can be used in the form PolynomialMapPreimage(..) 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][PolynomialMapPreimage](..).

Examples

 > $\mathrm{with}\left(\mathrm{RegularChains}\right):$
 > $\mathrm{with}\left(\mathrm{ConstructibleSetTools}\right):$
 > $R≔\mathrm{PolynomialRing}\left(\left[x,y,z\right]\right)$
 ${R}{≔}{\mathrm{polynomial_ring}}$ (1)
 > $S≔\mathrm{PolynomialRing}\left(\left[s,t\right]\right)$
 ${S}{≔}{\mathrm{polynomial_ring}}$ (2)

Note that the polynomial map should be a list of polynomials of R. Also the number of polynomials in PM equals the number of variables of S.

 > $\mathrm{MP}≔\left[{x}^{2},{y}^{2}\right]$
 ${\mathrm{MP}}{≔}\left[{{x}}^{{2}}{,}{{y}}^{{2}}\right]$ (3)
 > $F≔\left[s-1,t-1\right]$
 ${F}{≔}\left[{s}{-}{1}{,}{t}{-}{1}\right]$ (4)
 > $\mathrm{cs}≔\mathrm{PolynomialMapPreimage}\left(F,\mathrm{MP},R,S\right)$
 ${\mathrm{cs}}{≔}{\mathrm{constructible_set}}$ (5)
 > $\mathrm{Info}\left(\mathrm{cs},R\right)$
 $\left[\left[{x}{+}{1}{,}{y}{-}{1}\right]{,}\left[{1}\right]\right]{,}\left[\left[{x}{-}{1}{,}{y}{-}{1}\right]{,}\left[{1}\right]\right]{,}\left[\left[{x}{+}{1}{,}{y}{+}{1}\right]{,}\left[{1}\right]\right]{,}\left[\left[{x}{-}{1}{,}{y}{+}{1}\right]{,}\left[{1}\right]\right]$ (6)