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RegularChains[SemiAlgebraicSetTools]

  

RepresentingQuantifierFreeFormula

  

return the quantifier-free formula of a parametric box or a regular semi-algebraic system

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

RepresentingQuantifierFreeFormula(pbx)

RepresentingQuantifierFreeFormula(rst, R)

Parameters

pbx

-

a parametric box

rsas

-

a regular semi-algebraic system

R

-

a polynomial ring

Description

• 

The command RepresentingQuantifierFreeFormula(pbx) returns the representing quantifier-free formula of  the parametric box pbx.

• 

The command RepresentingQuantifierFreeFormula(rsas, R) returns the representing quantifier-free formula of the regular semi-algebraic system rsas.

• 

See the page SemiAlgebraicSetTools for the definition of a regular semi-algebraic system and that of a parametric box.

Examples

withRegularChains:

withParametricSystemTools:

withSemiAlgebraicSetTools:

RPolynomialRingx,b,a,c

R:=polynomial_ring

(1)

Fax2+bx+c

F:=ax2+bx+c

(2)

N

N:=

(3)

Px

P:=x

(4)

Ha

H:=a

(5)

rrcRealRootClassificationF,,x,a,3,2,R

rrc:=regular_semi_algebraic_set,border_polynomial

(6)

rsasrrc11

rsas:=regular_semi_algebraic_set

(7)

pbxRepresentingBoxrsas,R

pbx:=parametric_box

(8)

IsParametricBoxpbx

true

(9)

qffRepresentingQuantifierFreeFormulapbx

qff:=quantifier_free_formula

(10)

Infoqff,R

c,a,b,4acb2,1,1,1,1,1,1,1,1

(11)

Fax2&plus;bx&plus;c&equals;0&comma;0<x&comma;a0

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

(12)

RPolynomialRingx&comma;c&comma;b&comma;a

R:=polynomial_ring

(13)

outLazyRealTriangularizeF&comma;R&comma;output&equals;list

out:=regular_semi_algebraic_system

(14)

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

(15)

PPositiveInequalitiesout1&comma;R

P:=x

(16)

rcRepresentingChainout1&comma;R&semi;Displayrc&comma;R

rc:=regular_chain

&lcub;ax2&plus;bx&plus;c&equals;0a0

(17)

qffRepresentingQuantifierFreeFormulaout1&semi;Displayqff&comma;R

qff:=quantifier_free_formula

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

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

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

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

(18)

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

(19)

See Also

LazyRealTriangularize

PositiveInequalities

RealRootClassification

RealTriangularize

RegularChains

RepresentingChain

RepresentingRootIndex

 


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