RegularChains[ConstructibleSetTools] - Maple Help

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RegularChains[ConstructibleSetTools]

 RationalMapImage
 compute the image of a variety or a constructible set under a rational map

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

Parameters

 F - list of polynomials 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 RationalMapImage(F, RM, R, S) returns a constructible set cs which is the image 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 RationalMapImage(F, H, RM, R, S) returns the image of the constructible set $V$-$W$ under the rational map RM.
 • The command RationalMapImage(CS, RM, R, S) returns the image 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 polynomials in RM is equal to the number of variables of ring S.

Examples

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

The following example is related to the tacnode curve.

 > $S≔\mathrm{PolynomialRing}\left(\left[t\right]\right)$
 ${S}{:=}{\mathrm{polynomial_ring}}$ (1)
 > $T≔\mathrm{PolynomialRing}\left(\left[x,y\right]\right)$
 ${T}{:=}{\mathrm{polynomial_ring}}$ (2)
 > $F≔\left[\right]$
 ${F}{:=}\left[{}\right]$ (3)
 > $\mathrm{RM}≔\left[\frac{{t}^{3}-6{t}^{2}+9t-2}{2{t}^{4}-16{t}^{3}+40{t}^{2}-32t+9},\frac{{t}^{2}-4t+4}{2{t}^{4}-16{t}^{3}+40{t}^{2}-32t+9}\right]$
 ${\mathrm{RM}}{:=}\left[\frac{{{t}}^{{3}}{-}{6}{}{{t}}^{{2}}{+}{9}{}{t}{-}{2}}{{2}{}{{t}}^{{4}}{-}{16}{}{{t}}^{{3}}{+}{40}{}{{t}}^{{2}}{-}{32}{}{t}{+}{9}}{,}\frac{{{t}}^{{2}}{-}{4}{}{t}{+}{4}}{{2}{}{{t}}^{{4}}{-}{16}{}{{t}}^{{3}}{+}{40}{}{{t}}^{{2}}{-}{32}{}{t}{+}{9}}\right]$ (4)
 > $\mathrm{cs}≔\mathrm{RationalMapImage}\left(F,\mathrm{RM},S,T\right)$
 ${\mathrm{cs}}{:=}{\mathrm{constructible_set}}$ (5)
 > $\mathrm{Info}\left(\mathrm{cs},T\right)$
 $\left[\left[{2}{}{{x}}^{{4}}{-}{3}{}{y}{}{{x}}^{{2}}{+}{{y}}^{{4}}{-}{2}{}{{y}}^{{3}}{+}{{y}}^{{2}}\right]{,}\left[{y}{,}\left({10}{}{y}{+}{2}\right){}{{x}}^{{2}}{+}{2}{}{{y}}^{{3}}{-}{{y}}^{{2}}{-}{y}{,}\left({964}{}{{y}}^{{6}}{-}{480}{}{{y}}^{{5}}{-}{6858}{}{{y}}^{{4}}{-}{4328}{}{{y}}^{{3}}{-}{888}{}{{y}}^{{2}}{-}{72}{}{y}{-}{2}\right){}{{x}}^{{2}}{-}{88}{}{{y}}^{{8}}{+}{2104}{}{{y}}^{{7}}{-}{2316}{}{{y}}^{{6}}{-}{943}{}{{y}}^{{5}}{+}{892}{}{{y}}^{{4}}{+}{318}{}{{y}}^{{3}}{+}{32}{}{{y}}^{{2}}{+}{y}\right]\right]{,}\left[\left[{x}{,}{y}{-}{1}\right]{,}\left[{1}\right]\right]{,}\left[\left[{x}{,}{y}\right]{,}\left[{1}\right]\right]$
 $\left\{\left[\left[{2}{}{{x}}^{{4}}{-}{3}{}{y}{}{{x}}^{{2}}{+}{{y}}^{{4}}{-}{2}{}{{y}}^{{3}}{+}{{y}}^{{2}}\right]{,}\left[{y}{,}\left({10}{}{y}{+}{2}\right){}{{x}}^{{2}}{+}{2}{}{{y}}^{{3}}{-}{{y}}^{{2}}{-}{y}{,}\left({964}{}{{y}}^{{6}}{-}{480}{}{{y}}^{{5}}{-}{6858}{}{{y}}^{{4}}{-}{4328}{}{{y}}^{{3}}{-}{888}{}{{y}}^{{2}}{-}{72}{}{y}{-}{2}\right){}{{x}}^{{2}}{-}{88}{}{{y}}^{{8}}{+}{2104}{}{{y}}^{{7}}{-}{2316}{}{{y}}^{{6}}{-}{943}{}{{y}}^{{5}}{+}{892}{}{{y}}^{{4}}{+}{318}{}{{y}}^{{3}}{+}{32}{}{{y}}^{{2}}{+}{y}\right]\right]{,}\left[\left[{x}{,}{y}\right]{,}\left[{1}\right]\right]{,}\left[\left[{x}{,}{y}{-}{1}\right]{,}\left[{1}\right]\right]\right\}$ (6)