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

withRegularChains:

withSemiAlgebraicSetTools:

F:=ax2&plus;bx&plus;c&equals;0&comma;0<x&comma;a0

F:=ax2&plus;bx&plus;c&equals;0&comma;0<x&comma;a0

(1)

R:=PolynomialRingx&comma;c&comma;b&comma;a

R:=polynomial_ring

(2)

out:=LazyRealTriangularizeF&comma;R&comma;output&equals;list

out:=regular_semi_algebraic_system

(3)

mapDisplay&comma;out&comma;R

&lcub;ax2+bx+c=0x>0&lcub;4ca+b2>0andb<0andc>0anda0or4ca+b2>0andb>0andc>0anda<0or4ca+b2>0andb>0andc<0anda0or4ca+b2>0andb<0andc<0anda>0

(4)

P:=PositiveInequalitiesout1&comma;R

P:=x

(5)

rc:=RepresentingChainout1&comma;R

rc:=regular_chain

(6)

qff:=RepresentingQuantifierFreeFormulaout1

qff:=quantifier_free_formula

(7)

Displayqff&comma;R

4ca+b2>0andb<0andc>0anda0

or4ca+b2>0andb>0andc>0anda<0

or4ca+b2>0andb>0andc<0anda0

or4ca+b2>0andb<0andc<0anda>0

(8)

Displayout1&comma;R

&lcub;ax2+bx+c=0x>0&lcub;4ca+b2>0andb<0andc>0anda0or4ca+b2>0andb>0andc>0anda<0or4ca+b2>0andb>0andc<0anda0or4ca+b2>0andb<0andc<0anda>0

(9)

See Also

RealTriangularize, RegularChains, RepresentingChain, RepresentingQuantifierFreeFormula, SemiAlgebraicSetTools


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