PostitiveInequalities - Maple Help
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RegularChains[SemiAlgebraicSetTools]

 PositiveInequalities
 return the positive inequalities of a regular semi-algebraic system

 Calling Sequence PositiveInequalities(rsas, R)

Parameters

 rsas - regular semi-algebraic system R - polynomial ring

Description

 • The command PositiveInequalities(rsas, R) returns the defining positive inequalities of the regular semi-algebraic system rsas. The polynomials must belong to R which must have characteristic zero.
 See the page SemiAlgebraicSetTools for the definitions of a regular semi-algebraic system and that of a regular semi-algebraic set.

Examples

 > $\mathrm{with}\left(\mathrm{RegularChains}\right):$
 > $\mathrm{with}\left(\mathrm{SemiAlgebraicSetTools}\right):$
 > $F≔\left[a{x}^{2}+bx+c=0,0
 ${F}{≔}\left[{a}{}{{x}}^{{2}}{+}{b}{}{x}{+}{c}{=}{0}{,}{0}{<}{x}{,}{a}{\ne }{0}\right]$ (1)
 > $R≔\mathrm{PolynomialRing}\left(\left[x,c,b,a\right]\right)$
 ${R}{≔}{\mathrm{polynomial_ring}}$ (2)
 > $\mathrm{out}≔\mathrm{LazyRealTriangularize}\left(F,R,\mathrm{output}=\mathrm{list}\right)$
 ${\mathrm{out}}{≔}\left[{\mathrm{regular_semi_algebraic_system}}\right]$ (3)
 > $\mathrm{map}\left(\mathrm{Display},\mathrm{out},R\right)$
 $\left[\left\{\begin{array}{cc}{a}{}{{x}}^{{2}}{+}{b}{}{x}{+}{c}{=}{0}& {}\\ {x}{>}{0}& {}\\ \left\{\begin{array}{cc}{-}{4}{}{c}{}{a}{+}{{b}}^{{2}}{>}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{b}{<}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{c}{>}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{a}{\ne }{0}& {}\\ \phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{or}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{-}{4}{}{c}{}{a}{+}{{b}}^{{2}}{>}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{b}{>}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{c}{>}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{a}{<}{0}& {}\\ \phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{or}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{-}{4}{}{c}{}{a}{+}{{b}}^{{2}}{>}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{b}{>}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{c}{<}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{a}{\ne }{0}& {}\\ \phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{or}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{-}{4}{}{c}{}{a}{+}{{b}}^{{2}}{>}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{b}{<}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{c}{<}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{a}{>}{0}& {}\end{array}\right\& {}\end{array}\right\\right]$ (4)
 > $P≔\mathrm{PositiveInequalities}\left({\mathrm{out}}_{1},R\right)$
 ${P}{≔}\left[{x}\right]$ (5)
 > $\mathrm{rc}≔\mathrm{RepresentingChain}\left({\mathrm{out}}_{1},R\right)$
 ${\mathrm{rc}}{≔}{\mathrm{regular_chain}}$ (6)
 > $\mathrm{qff}≔\mathrm{RepresentingQuantifierFreeFormula}\left({\mathrm{out}}_{1}\right)$
 ${\mathrm{qff}}{≔}{\mathrm{quantifier_free_formula}}$ (7)
 > $\mathrm{Display}\left(\mathrm{qff},R\right)$
 ${-}{4}{}{c}{}{a}{+}{{b}}^{{2}}{>}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{b}{<}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{c}{>}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{a}{\ne }{0}$
 $\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{or}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{-}{4}{}{c}{}{a}{+}{{b}}^{{2}}{>}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{b}{>}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{c}{>}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{a}{<}{0}$
 $\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{or}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{-}{4}{}{c}{}{a}{+}{{b}}^{{2}}{>}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{b}{>}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{c}{<}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{a}{\ne }{0}$
 $\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{or}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{-}{4}{}{c}{}{a}{+}{{b}}^{{2}}{>}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{b}{<}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{c}{<}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{a}{>}{0}$ (8)
 > $\mathrm{Display}\left({\mathrm{out}}_{1},R\right)$
 $\left\{\begin{array}{cc}{a}{}{{x}}^{{2}}{+}{b}{}{x}{+}{c}{=}{0}& {}\\ {x}{>}{0}& {}\\ \left\{\begin{array}{cc}{-}{4}{}{c}{}{a}{+}{{b}}^{{2}}{>}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{b}{<}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{c}{>}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{a}{\ne }{0}& {}\\ \phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{or}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{-}{4}{}{c}{}{a}{+}{{b}}^{{2}}{>}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{b}{>}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{c}{>}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{a}{<}{0}& {}\\ \phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{or}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{-}{4}{}{c}{}{a}{+}{{b}}^{{2}}{>}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{b}{>}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{c}{<}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{a}{\ne }{0}& {}\\ \phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{or}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{-}{4}{}{c}{}{a}{+}{{b}}^{{2}}{>}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{b}{<}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{c}{<}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{a}{>}{0}& {}\end{array}\right\& {}\end{array}\right\$ (9)

Compatibility

 • The RegularChains[SemiAlgebraicSetTools][PositiveInequalities] command was introduced in Maple 15.
 • For more information on Maple 15 changes, see Updates in Maple 15.