RegularChains[ConstructibleSetTools] - Maple Programming Help

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RegularChains[ConstructibleSetTools]

 RationalMapPreimage
 compute the preimage of a variety under a polynomial map

 Calling Sequence RationalMapPreimage(F, RM, R, S) RationalMapPreimage(F, H, RM, R, S) RationalMapPreimage(CS, RM, R, S)

Parameters

 F - list of polynomials of S RM - a list of rational functions in R R - a polynomial ring (source) S - a polynomial ring (target) H - list of polynomials CS - constructible set

Description

 • The command RationalMapPreimage(F, RM, R, S) returns a constructible set cs over R. cs is the preimage of the variety $V\left(F\right)$ under the rational map RM.
 • If H is specified, let $W$ be the variety defined by the product of polynomials in H. The command RationalMapPreimage(F, H, RM, R, S) returns the preimage of the constructible set $V$-$W$ under the rational map RM.
 • The command RationalMapPreimage(CS, RM, R, S) returns the preimage of the constructible set CS under the rational map RM.
 • Both rings R and S should be over the same ground field.
 • The variable sets of R and S should be disjoint.
 • The number of rational functions in RM is equal to the number of variables of ring S.

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 rational map should be a list of rational functions of R. Also, the number of polynomials in RM equals the number of variables of S.

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