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.

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

$\mathrm{with}\left(\mathrm{SemiAlgebraicSetTools}\right)\:$

$F≔\left[a{x}^2\&plus;bx\&plus;c\&equals;0\&comma;0<x\&comma;a\ne 0\right]$

$F≔\left[a{x}^2+bx+c=0\&comma;0<x\&comma;a\ne 0\right]$

$R≔\mathrm{PolynomialRing}\left(\left[x\&comma;c\&comma;b\&comma;a\right]\right)$

$R≔\mathrm{polynomial\_ring}$

$\mathrm{out}≔\mathrm{LazyRealTriangularize}\left(F\&comma;R\&comma;\mathrm{output}\&equals;\mathrm{list}\right)$

$\mathrm{out}≔\left[\mathrm{regular\_semi\_algebraic\_system}\right]$

$\mathrm{map}\left(\mathrm{Display}\&comma;\mathrm{out}\&comma;R\right)$

$\left[\left\{\begin{array}{cc}a{x}^2+bx+c=0& \\ x>0& \\ \left\{\begin{array}{cc}-4ca+b^2>0\mathbf{and}b<0\mathbf{and}c>0\mathbf{and}a\ne 0& \\ \mathbf{or}-4ca+b^2>0\mathbf{and}b>0\mathbf{and}c>0\mathbf{and}a<0& \\ \mathbf{or}-4ca+b^2>0\mathbf{and}b>0\mathbf{and}c<0\mathbf{and}a\ne 0& \\ \mathbf{or}-4ca+b^2>0\mathbf{and}b<0\mathbf{and}c<0\mathbf{and}a>0& \end{array}\right.& \end{array}\right.\right]$

$P≔\mathrm{PositiveInequalities}\left({\mathrm{out}}_1\&comma;R\right)$

$P≔\left[x\right]$

$\mathrm{rc}≔\mathrm{RepresentingChain}\left({\mathrm{out}}_1\&comma;R\right)$

$\mathrm{rc}≔\mathrm{regular\_chain}$

$\mathrm{qff}≔\mathrm{RepresentingQuantifierFreeFormula}\left({\mathrm{out}}_1\right)$

$\mathrm{qff}≔\mathrm{quantifier\_free\_formula}$

$\mathrm{Display}\left(\mathrm{qff}\&comma;R\right)$

$-4ca+b^2>0\mathbf{and}b<0\mathbf{and}c>0\mathbf{and}a\ne 0$

$\mathbf{or}-4ca+b^2>0\mathbf{and}b>0\mathbf{and}c>0\mathbf{and}a<0$

$\mathbf{or}-4ca+b^2>0\mathbf{and}b>0\mathbf{and}c<0\mathbf{and}a\ne 0$

$\mathbf{or}-4ca+b^2>0\mathbf{and}b<0\mathbf{and}c<0\mathbf{and}a>0$

$\mathrm{Display}\left({\mathrm{out}}_1\&comma;R\right)$

$\left\{\begin{array}{cc}a{x}^2+bx+c=0& \\ x>0& \\ \left\{\begin{array}{cc}-4ca+b^2>0\mathbf{and}b<0\mathbf{and}c>0\mathbf{and}a\ne 0& \\ \mathbf{or}-4ca+b^2>0\mathbf{and}b>0\mathbf{and}c>0\mathbf{and}a<0& \\ \mathbf{or}-4ca+b^2>0\mathbf{and}b>0\mathbf{and}c<0\mathbf{and}a\ne 0& \\ \mathbf{or}-4ca+b^2>0\mathbf{and}b<0\mathbf{and}c<0\mathbf{and}a>0& \end{array}\right.& \end{array}\right.$

The RegularChains[SemiAlgebraicSetTools][PositiveInequalities] command was introduced in Maple 15.

For more information on Maple 15 changes, see Updates in Maple 15.