 Inequations - Maple Help

RegularChains

 Inequations
 list of inequations of the regular chain Calling Sequence Inequations(rc, R) Parameters

 rc - regular chain of R R - polynomial ring Description

 • The command Inequations(rc,R) returns the set of the initials of rc.
 • By definition, a zero of the regular chain rc is a common zero of its equations that does not cancel any of the initials of rc.
 • This command is part of the RegularChains package, so it can be used in the form Inequations(..) only after executing the command with(RegularChains).  However, it can always be accessed through the long form of the command by using RegularChains[Inequations](..). Examples

 > $\mathrm{with}\left(\mathrm{RegularChains}\right):$
 > $R≔\mathrm{PolynomialRing}\left(\left[x,y,a,b,c,d,g,h\right]\right)$
 ${R}{≔}{\mathrm{polynomial_ring}}$ (1)
 > $\mathrm{sys}≔\left\{ax+by-g,cx+dy-h\right\}$
 ${\mathrm{sys}}{≔}\left\{{a}{}{x}{+}{b}{}{y}{-}{g}{,}{c}{}{x}{+}{d}{}{y}{-}{h}\right\}$ (2)

First, we compute the generic solutions of sys, that is a triangular decomposition of the zeros of sys in the sense of Kalkbrener.

 > $\mathrm{dec}≔\mathrm{Triangularize}\left(\mathrm{sys},R\right);$$\mathrm{map}\left(\mathrm{Equations},\mathrm{dec},R\right)$
 ${\mathrm{dec}}{≔}\left[{\mathrm{regular_chain}}\right]$
 $\left[\left[{c}{}{x}{+}{y}{}{d}{-}{h}{,}\left({a}{}{d}{-}{b}{}{c}\right){}{y}{-}{a}{}{h}{+}{c}{}{g}\right]\right]$ (3)

Then we compute all the solutions (generic or not), that is a triangular decomposition in the sense of Lazard. For each computed regular chain, we show its equations and inequations.

 > $\mathrm{dec}≔\mathrm{Triangularize}\left(\mathrm{sys},R,\mathrm{output}=\mathrm{lazard}\right)$
 ${\mathrm{dec}}{≔}\left[{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}\right]$ (4)
 > $\mathrm{map}\left(\mathrm{Equations},\mathrm{dec},R\right)$
 $\left[\left[{c}{}{x}{+}{d}{}{y}{-}{h}{,}\left({d}{}{a}{-}{b}{}{c}\right){}{y}{-}{h}{}{a}{+}{c}{}{g}\right]{,}\left[{c}{}{x}{+}{d}{}{y}{-}{h}{,}{d}{}{a}{-}{b}{}{c}{,}{h}{}{b}{-}{d}{}{g}\right]{,}\left[{a}{}{x}{+}{b}{}{y}{-}{g}{,}{d}{}{y}{-}{h}{,}{c}\right]{,}\left[{d}{}{y}{-}{h}{,}{a}{,}{h}{}{b}{-}{d}{}{g}{,}{c}\right]{,}\left[{c}{}{x}{-}{h}{,}{h}{}{a}{-}{c}{}{g}{,}{b}{,}{d}\right]{,}\left[{a}{}{x}{+}{b}{}{y}{-}{g}{,}{c}{,}{d}{,}{h}\right]{,}\left[{c}{}{x}{+}{d}{}{y}{,}{d}{}{a}{-}{b}{}{c}{,}{g}{,}{h}\right]{,}\left[{b}{}{y}{-}{g}{,}{a}{,}{c}{,}{d}{,}{h}\right]{,}\left[{y}{,}{a}{,}{c}{,}{g}{,}{h}\right]{,}\left[{x}{,}{b}{,}{d}{,}{g}{,}{h}\right]{,}\left[{a}{,}{b}{,}{c}{,}{d}{,}{g}{,}{h}\right]\right]$ (5)
 > $\mathrm{map}\left(\mathrm{Inequations},\mathrm{dec},R\right)$
 $\left[\left\{{c}{,}{d}{}{a}{-}{b}{}{c}\right\}{,}\left\{{c}{,}{d}{,}{h}\right\}{,}\left\{{a}{,}{d}\right\}{,}\left\{{d}{,}{h}\right\}{,}\left\{{c}{,}{h}\right\}{,}\left\{{a}\right\}{,}\left\{{c}{,}{d}\right\}{,}\left\{{b}\right\}{,}{\varnothing }{,}{\varnothing }{,}{\varnothing }\right]$ (6)
 > $\left[\mathrm{seq}\left(\left[\mathrm{eq}=\mathrm{Equations}\left(\mathrm{dec}\left[i\right],R\right),\mathrm{ineq}=\mathrm{Inequations}\left(\mathrm{dec}\left[i\right],R\right)\right],i=1..\mathrm{nops}\left(\mathrm{dec}\right)\right)\right]$
 $\left[\left[{\mathrm{eq}}{=}\left[{c}{}{x}{+}{d}{}{y}{-}{h}{,}\left({d}{}{a}{-}{b}{}{c}\right){}{y}{-}{h}{}{a}{+}{c}{}{g}\right]{,}{\mathrm{ineq}}{=}\left\{{c}{,}{d}{}{a}{-}{b}{}{c}\right\}\right]{,}\left[{\mathrm{eq}}{=}\left[{c}{}{x}{+}{d}{}{y}{-}{h}{,}{d}{}{a}{-}{b}{}{c}{,}{h}{}{b}{-}{d}{}{g}\right]{,}{\mathrm{ineq}}{=}\left\{{c}{,}{d}{,}{h}\right\}\right]{,}\left[{\mathrm{eq}}{=}\left[{a}{}{x}{+}{b}{}{y}{-}{g}{,}{d}{}{y}{-}{h}{,}{c}\right]{,}{\mathrm{ineq}}{=}\left\{{a}{,}{d}\right\}\right]{,}\left[{\mathrm{eq}}{=}\left[{d}{}{y}{-}{h}{,}{a}{,}{h}{}{b}{-}{d}{}{g}{,}{c}\right]{,}{\mathrm{ineq}}{=}\left\{{d}{,}{h}\right\}\right]{,}\left[{\mathrm{eq}}{=}\left[{c}{}{x}{-}{h}{,}{h}{}{a}{-}{c}{}{g}{,}{b}{,}{d}\right]{,}{\mathrm{ineq}}{=}\left\{{c}{,}{h}\right\}\right]{,}\left[{\mathrm{eq}}{=}\left[{a}{}{x}{+}{b}{}{y}{-}{g}{,}{c}{,}{d}{,}{h}\right]{,}{\mathrm{ineq}}{=}\left\{{a}\right\}\right]{,}\left[{\mathrm{eq}}{=}\left[{c}{}{x}{+}{d}{}{y}{,}{d}{}{a}{-}{b}{}{c}{,}{g}{,}{h}\right]{,}{\mathrm{ineq}}{=}\left\{{c}{,}{d}\right\}\right]{,}\left[{\mathrm{eq}}{=}\left[{b}{}{y}{-}{g}{,}{a}{,}{c}{,}{d}{,}{h}\right]{,}{\mathrm{ineq}}{=}\left\{{b}\right\}\right]{,}\left[{\mathrm{eq}}{=}\left[{y}{,}{a}{,}{c}{,}{g}{,}{h}\right]{,}{\mathrm{ineq}}{=}{\varnothing }\right]{,}\left[{\mathrm{eq}}{=}\left[{x}{,}{b}{,}{d}{,}{g}{,}{h}\right]{,}{\mathrm{ineq}}{=}{\varnothing }\right]{,}\left[{\mathrm{eq}}{=}\left[{a}{,}{b}{,}{c}{,}{d}{,}{g}{,}{h}\right]{,}{\mathrm{ineq}}{=}{\varnothing }\right]\right]$ (7)