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RegularChains

 Info
 the defining polynomials of a structure in a raw format
 Display
 display the defining polynomials of a structure in a pretty format

 Calling Sequence Info(rc, R) Info(rs, R) Info(cs, R) Info(cadcell, R) Info(rsaset, R) Info(saset, R) Info(rsasys, R) Info(ssasys, R) Info(qff, R) Info(b, R) Info(pb, R) Info(bp, R) Info(src, R) Display(rc, R) Display(rs, R) Display(cs, R) Display(cadcell, R) Display(rsaset, R) Display(saset, R) Display(rsasys, R) Display(ssasys, R) Display(qff, R) Display(b, R) Display(pb, R) Display(bp, R) Display(src, R)

Parameters

 rc - regular chain rs - regular system cs - constructible set cadcell - CAD cell rsaset - regular semi-algebraic set saset - semi-algebraic set rsasys - regular semi-algebraic system ssasys - squarefree semi-algebraic system qff - quantifier-free formula b - box pb - parametric box bp - border polynomial src - subresultant chain R - polynomial ring

Description

 • If the first parameter is a regular chain rc of R, Info prints the defining equations of rc while Display prints the system of equations and inequations whose solution set is the quasi-component of rc.
 • If the first parameter is a regular system rs of R, Info and Display prints the defining equations and inequations of rs whose solution set is the zero-set of rs.
 • If the first parameter is a constructible set cs of R, Info and Display print the defining regular systems of cs.
 • If the first parameter is a regular semi-algebraic system rsasys of R, Info and Display print the defining regular chain, positive inequalities and quantifier-free formula of rsasys.
 • If the first parameter is a squarefree semi-algebraic system ssasys of R, Info and Display print the defining regular chain and positive inequalities of ssasys.
 • If the first parameter is a regular semi-algebraic set rsaset, then Info and Display print its defining regular chain, root index list and quantifier-free formula.
 • If the first parameter is a semi-algebraic set saset, then Info and Display print the defining regular semi-algebraic sets of saset.
 • If the first parameter is a quantifier-free formula qff, then Info and Display print its defining inequations and inequalities.
 • If the first parameter is a CAD cell cadcell, then Info and Display print its defining inequations and inequalities.
 • If the first parameter is a box b , then Info and Display print isolation interval for each of the coordinates of the real point isolated by b.
 • If the first parameter is a parametric box pb, then Info and Display print its defining regular chain, root index list and quantifier-free formula.
 • If the first parameter is a border polynomial bp, then Info and Display print its defining polynomial factors.
 • If the first parameter is a subresultant chain src, then Info prints its defining subresultant polynomials by ascending index while Display prints them in descending index.
 • If the first parameter is a list, then both Info and Display are applied to all list elements recursively.
 • The command Info is part of the RegularChains package, so it can be used in the form Info(..) only after executing the command with(RegularChains).  However, it can always be accessed through the long form of the command by using RegularChains[Info](..). This is also true for Display.

Examples

 > $\mathrm{with}\left(\mathrm{RegularChains}\right):$
 > $\mathrm{with}\left(\mathrm{ConstructibleSetTools}\right):$$\mathrm{with}\left(\mathrm{ParametricSystemTools}\right):$$\mathrm{with}\left(\mathrm{SemiAlgebraicSetTools}\right):$$\mathrm{with}\left(\mathrm{ChainTools}\right):$
 > $R≔\mathrm{PolynomialRing}\left(\left[x,y,s\right]\right);$$F≔\left[s-\left(y+1\right)x,s-\left(x+1\right)y\right]$
 ${R}{≔}{\mathrm{polynomial_ring}}$
 ${F}{≔}\left[{s}{-}\left({y}{+}{1}\right){}{x}{,}{s}{-}\left({x}{+}{1}\right){}{y}\right]$ (1)

Print the variables, parameters and characteristic of a polynomial ring.

 > $\mathrm{Display}\left(R\right)$
 $\left\{\begin{array}{lll}{\mathrm{Variables}}& {:}& {}\left[{x}{,}{y}{,}{s}\right]\\ {\mathrm{Parameters}}& {:}& {}{\varnothing }\\ {\mathrm{Characteristic}}& {:}& {}{0}\end{array}\right\$ (2)

Print the regular chains in a triangular decomposition of a system of polynomial equations.

 > $\mathrm{dec}≔\mathrm{Triangularize}\left(F,R,\mathrm{output}=\mathrm{lazard}\right)$
 ${\mathrm{dec}}{≔}\left[{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}\right]$ (3)
 > $\mathrm{Info}\left(\mathrm{dec},R\right)$
 $\left[\left[\left({y}{+}{1}\right){}{x}{-}{s}{,}{{y}}^{{2}}{+}{y}{-}{s}\right]{,}\left[{x}{+}{1}{,}{y}{+}{1}{,}{s}\right]\right]$ (4)
 > $\mathrm{Display}\left(\mathrm{dec},R\right)$
 $\left[\left\{\begin{array}{cc}\left({y}{+}{1}\right){}{x}{-}{s}{=}{0}& {}\\ {{y}}^{{2}}{+}{y}{-}{s}{=}{0}& {}\\ {y}{+}{1}{\ne }{0}& {}\end{array}\right\{,}\left\{\begin{array}{cc}{x}{+}{1}{=}{0}& {}\\ {y}{+}{1}{=}{0}& {}\\ {s}{=}{0}& {}\end{array}\right\\right]$ (5)

Print the constructible set in a triangular decomposition of a system of polynomial equations and inequations..

 > $\mathrm{dec}≔\mathrm{Triangularize}\left(F,\left[x\right],R,\mathrm{output}=\mathrm{lazard}\right)$
 ${\mathrm{dec}}{≔}{\mathrm{constructible_set}}$ (6)
 > $\mathrm{Info}\left(\mathrm{dec},R\right)$
 $\left[\left[\left({y}{+}{1}\right){}{x}{-}{s}{,}{{y}}^{{2}}{+}{y}{-}{s}\right]{,}\left[{s}\right]\right]{,}\left[\left[{x}{+}{1}{,}{y}{+}{1}{,}{s}\right]{,}\left[{1}\right]\right]$ (7)
 > $\mathrm{Display}\left(\mathrm{dec},R\right)$
 $\left\{\begin{array}{cc}\left({y}{+}{1}\right){}{x}{-}{s}{=}{0}& {}\\ {{y}}^{{2}}{+}{y}{-}{s}{=}{0}& {}\\ {y}{+}{1}{\ne }{0}& {}\\ {s}{\ne }{0}& {}\end{array}\right\{,}\left\{\begin{array}{cc}{x}{+}{1}{=}{0}& {}\\ {y}{+}{1}{=}{0}& {}\\ {s}{=}{0}& {}\end{array}\right\$ (8)

Print the objects returned by a call to the RealRootClassification command.

 > $\mathrm{rrc}≔\mathrm{RealRootClassification}\left(F,\left[\right],\left[\right],\left[\right],1,1..k,R\right)$
 ${\mathrm{rrc}}{≔}\left[\left[{\mathrm{regular_semi_algebraic_set}}\right]{,}{\mathrm{border_polynomial}}\right]$ (9)
 > $\mathrm{Info}\left(\mathrm{rrc},R\right)$
 $\left[\left[\left[\left[\left[{4}{}{s}{+}{1}\right]{,}\left[\left[{1}\right]\right]\right]{,}\left[\right]{,}\left[\right]\right]\right]{,}\left[{s}{,}{s}{+}\frac{{1}}{{4}}\right]\right]$ (10)
 > $\mathrm{Display}\left(\mathrm{rrc},R\right)$
 $\left[\left[{4}{}{s}{+}{1}{>}{0}\right]{,}\left[{s}{,}{s}{+}\frac{{1}}{{4}}\right]\right]$ (11)
 > $\mathrm{pbox}≔\mathrm{RepresentingBox}\left(\mathrm{rrc}\left[1\right]\left[1\right],R\right)$
 ${\mathrm{pbox}}{≔}{\mathrm{parametric_box}}$ (12)
 > $\mathrm{Info}\left(\mathrm{pbox},R\right)$
 $\left[\left[\left[{4}{}{s}{+}{1}\right]{,}\left[\left[{1}\right]\right]\right]{,}\left[\right]{,}\left[\right]\right]$ (13)
 > $\mathrm{Display}\left(\mathrm{pbox},R\right)$
 ${4}{}{s}{+}{1}{>}{0}$ (14)
 > $R≔\mathrm{PolynomialRing}\left(\left[x,d,c,b,a\right]\right)$
 ${R}{≔}{\mathrm{polynomial_ring}}$ (15)
 > $F≔\left[{x}^{2}+d,{d}^{2}+{c}^{2}-1,{c}^{2}+bc+{a}^{2}\right]$
 ${F}{≔}\left[{{x}}^{{2}}{+}{d}{,}{{c}}^{{2}}{+}{{d}}^{{2}}{-}{1}{,}{{a}}^{{2}}{+}{b}{}{c}{+}{{c}}^{{2}}\right]$ (16)
 > $\mathrm{rrc}≔\mathrm{RealRootClassification}\left(F,\left[\right],\left[\right],\left[\right],4,1..k,R\right)$
 ${\mathrm{rrc}}{≔}\left[\left[{\mathrm{regular_semi_algebraic_set}}\right]{,}{\mathrm{border_polynomial}}\right]$ (17)
 > $\mathrm{rsat}≔\mathrm{rrc}\left[1\right]\left[1\right]$
 ${\mathrm{rsat}}{≔}{\mathrm{regular_semi_algebraic_set}}$ (18)
 > $\mathrm{Info}\left(\mathrm{rsat},R\right)$
 $\left[\left[\left[{d}\right]{,}\left[\left[{-1}\right]\right]\right]{,}\left[{-}{{d}}^{{2}}{+}{{a}}^{{2}}{+}{c}{}{b}{+}{1}{,}{{d}}^{{2}}{+}{{c}}^{{2}}{-}{1}\right]{,}\left[\left[{1}{,}{1}\right]{,}\left[{1}{,}{2}\right]{,}\left[{2}{,}{1}\right]{,}\left[{2}{,}{2}\right]\right]\right]$ (19)
 > $\mathrm{Display}\left(\mathrm{rsat},R\right)$
 $\left\{\begin{array}{cc}{a}{=}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{c}{}{b}{-}{{d}}^{{2}}{+}{1}{,}{\mathrm{index}}{=}{{\mathrm{real}}}_{{1}}\right)& {}\\ {c}{=}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{{d}}^{{2}}{-}{1}{,}{\mathrm{index}}{=}{{\mathrm{real}}}_{{1}}\right)& {}\\ {d}{<}{0}& {}\end{array}\right\{,}\left\{\begin{array}{cc}{a}{=}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{c}{}{b}{-}{{d}}^{{2}}{+}{1}{,}{\mathrm{index}}{=}{{\mathrm{real}}}_{{2}}\right)& {}\\ {c}{=}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{{d}}^{{2}}{-}{1}{,}{\mathrm{index}}{=}{{\mathrm{real}}}_{{1}}\right)& {}\\ {d}{<}{0}& {}\end{array}\right\{,}\left\{\begin{array}{cc}{a}{=}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{c}{}{b}{-}{{d}}^{{2}}{+}{1}{,}{\mathrm{index}}{=}{{\mathrm{real}}}_{{1}}\right)& {}\\ {c}{=}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{{d}}^{{2}}{-}{1}{,}{\mathrm{index}}{=}{{\mathrm{real}}}_{{2}}\right)& {}\\ {d}{<}{0}& {}\end{array}\right\{,}\left\{\begin{array}{cc}{a}{=}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{c}{}{b}{-}{{d}}^{{2}}{+}{1}{,}{\mathrm{index}}{=}{{\mathrm{real}}}_{{2}}\right)& {}\\ {c}{=}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{{d}}^{{2}}{-}{1}{,}{\mathrm{index}}{=}{{\mathrm{real}}}_{{2}}\right)& {}\\ {d}{<}{0}& {}\end{array}\right\$ (20)

Print the regular semi-algebraic systems returned by a call  to the RealTriangularize command.

 > $R≔\mathrm{PolynomialRing}\left(\left[x,y,s\right]\right)$
 ${R}{≔}{\mathrm{polynomial_ring}}$ (21)
 > $F≔\left[s-\left(y+1\right)x,s-\left(x+1\right)y\right]$
 ${F}{≔}\left[{s}{-}\left({y}{+}{1}\right){}{x}{,}{s}{-}\left({x}{+}{1}\right){}{y}\right]$ (22)
 > $\mathrm{rtd}≔\mathrm{RealTriangularize}\left(F,R\right)$
 ${\mathrm{rtd}}{≔}\left[{\mathrm{regular_semi_algebraic_system}}{,}{\mathrm{regular_semi_algebraic_system}}{,}{\mathrm{regular_semi_algebraic_system}}{,}{\mathrm{regular_semi_algebraic_system}}\right]$ (23)
 > $\mathrm{Info}\left(\mathrm{rtd},R\right)$
 $\left[\left[\left[\left[{4}{}{s}{+}{1}{,}{s}\right]{,}\left[\left[{1}{,}{0}\right]\right]\right]{,}\left[\left({y}{+}{1}\right){}{x}{-}{s}{,}{{y}}^{{2}}{+}{y}{-}{s}\right]{,}\left[\right]\right]{,}\left[{\mathrm{true}}{,}\left[{x}{,}{y}{,}{s}\right]{,}\left[\right]\right]{,}\left[{\mathrm{true}}{,}\left[{2}{}{x}{+}{1}{,}{2}{}{y}{+}{1}{,}{4}{}{s}{+}{1}\right]{,}\left[\right]\right]{,}\left[{\mathrm{true}}{,}\left[{x}{+}{1}{,}{y}{+}{1}{,}{s}\right]{,}\left[\right]\right]\right]$ (24)
 > $\mathrm{Display}\left(\mathrm{rtd},R\right)$
 $\left[\left\{\begin{array}{cc}\left({y}{+}{1}\right){}{x}{-}{s}{=}{0}& {}\\ {{y}}^{{2}}{+}{y}{-}{s}{=}{0}& {}\\ {4}{}{s}{+}{1}{>}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{s}{\ne }{0}& {}\end{array}\right\{,}\left\{\begin{array}{cc}{x}{=}{0}& {}\\ {y}{=}{0}& {}\\ {s}{=}{0}& {}\end{array}\right\{,}\left\{\begin{array}{cc}{2}{}{x}{+}{1}{=}{0}& {}\\ {2}{}{y}{+}{1}{=}{0}& {}\\ {4}{}{s}{+}{1}{=}{0}& {}\end{array}\right\{,}\left\{\begin{array}{cc}{x}{+}{1}{=}{0}& {}\\ {y}{+}{1}{=}{0}& {}\\ {s}{=}{0}& {}\end{array}\right\\right]$ (25)

Print the isolation boxes returned by a call to the RealRootIsolate command.

 > $R≔\mathrm{PolynomialRing}\left(\left[y,x\right]\right)$
 ${R}{≔}{\mathrm{polynomial_ring}}$ (26)
 > $F≔\left[1-\left(y+1\right)x,1-\left(x+1\right)y\right]$
 ${F}{≔}\left[{1}{-}\left({y}{+}{1}\right){}{x}{,}{1}{-}\left({x}{+}{1}\right){}{y}\right]$ (27)
 > $\mathrm{rri}≔\mathrm{RealRootIsolate}\left(F,\left[\right],\left[\right],\left[\right],R\right)$
 ${\mathrm{rri}}{≔}\left[{\mathrm{box}}{,}{\mathrm{box}}\right]$ (28)
 > $\mathrm{Info}\left(\mathrm{rri},R\right)$
 $\left[\left[\left[{y}{=}\left[{-}\frac{{13}}{{8}}{,}{-}\frac{{3}}{{2}}\right]{,}{x}{=}\left[{-}\frac{{13}}{{8}}{,}{-}\frac{{207}}{{128}}\right]\right]{,}\left[\left({x}{+}{1}\right){}{y}{-}{1}{,}{{x}}^{{2}}{+}{x}{-}{1}\right]\right]{,}\left[\left[{y}{=}\left[\frac{{1}}{{2}}{,}\frac{{5}}{{8}}\right]{,}{x}{=}\left[\frac{{79}}{{128}}{,}\frac{{5}}{{8}}\right]\right]{,}\left[\left({x}{+}{1}\right){}{y}{-}{1}{,}{{x}}^{{2}}{+}{x}{-}{1}\right]\right]\right]$ (29)
 > $\mathrm{Display}\left(\mathrm{rri},R\right)$
 $\left[\left\{\begin{array}{cc}{y}{=}\left[{-}\frac{{13}}{{8}}{,}{-}\frac{{3}}{{2}}\right]& {}\\ {x}{=}\left[{-}\frac{{13}}{{8}}{,}{-}\frac{{207}}{{128}}\right]& {}\end{array}\right\{,}\left\{\begin{array}{cc}{y}{=}\left[\frac{{1}}{{2}}{,}\frac{{5}}{{8}}\right]& {}\\ {x}{=}\left[\frac{{79}}{{128}}{,}\frac{{5}}{{8}}\right]& {}\end{array}\right\\right]$ (30)

Print the polynomials of a subresultant chain.

 > $R≔\mathrm{PolynomialRing}\left(\left[y,x,s\right]\right)$
 ${R}{≔}{\mathrm{polynomial_ring}}$ (31)
 > $p≔s-\left({y}^{3}+1\right)x$
 ${p}{≔}{s}{-}\left({{y}}^{{3}}{+}{1}\right){}{x}$ (32)
 > $q≔s-\left(x+1\right){y}^{2}$
 ${q}{≔}{s}{-}\left({x}{+}{1}\right){}{{y}}^{{2}}$ (33)
 > $\mathrm{src}≔\mathrm{SubresultantChain}\left(p,q,y,R\right)$
 ${\mathrm{src}}{≔}{\mathrm{subresultant_chain}}$ (34)
 > $\mathrm{Info}\left(\mathrm{src},R\right)$
 $\left[{{s}}^{{3}}{}{{x}}^{{2}}{-}{{s}}^{{2}}{}{{x}}^{{3}}{+}{2}{}{{x}}^{{4}}{}{s}{-}{{x}}^{{5}}{-}{3}{}{{s}}^{{2}}{}{{x}}^{{2}}{+}{6}{}{s}{}{{x}}^{{3}}{-}{3}{}{{x}}^{{4}}{-}{3}{}{{s}}^{{2}}{}{x}{+}{6}{}{s}{}{{x}}^{{2}}{-}{3}{}{{x}}^{{3}}{-}{{s}}^{{2}}{+}{2}{}{x}{}{s}{-}{{x}}^{{2}}{,}{-}{s}{}{{x}}^{{2}}{}{y}{+}{s}{}{{x}}^{{2}}{-}{y}{}{x}{}{s}{-}{{x}}^{{3}}{+}{2}{}{x}{}{s}{-}{2}{}{{x}}^{{2}}{+}{s}{-}{x}\right]$ (35)
 > $\mathrm{Display}\left(\mathrm{src},R\right)$
 $\left\{\begin{array}{cc}\left({-}{s}{}{{x}}^{{2}}{-}{x}{}{s}\right){}{y}{-}{{x}}^{{3}}{+}\left({s}{-}{2}\right){}{{x}}^{{2}}{+}\left({2}{}{s}{-}{1}\right){}{x}{+}{s}& {}\\ {-}{{x}}^{{5}}{+}\left({2}{}{s}{-}{3}\right){}{{x}}^{{4}}{+}\left({-}{{s}}^{{2}}{+}{6}{}{s}{-}{3}\right){}{{x}}^{{3}}{+}\left({{s}}^{{3}}{-}{3}{}{{s}}^{{2}}{+}{6}{}{s}{-}{1}\right){}{{x}}^{{2}}{+}\left({-}{3}{}{{s}}^{{2}}{+}{2}{}{s}\right){}{x}{-}{{s}}^{{2}}& {}\end{array}\right\$ (36)

Compatibility

 • The RegularChains[Display] command was introduced in Maple 15.