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Home : Support : Online Help : Mathematics : Factorization and Solving Equations : RegularChains : SemiAlgebraicSetTools Subpackage : RegularChains/ChainTools/RemoveRedundantComponents

RegularChains

  

ChainTools[RemoveRedundantComponents]

  

remove redundant quasi-components from a list of regular chains

  

SemiAlgebraicSetTools[RemoveRedundantComponents]

  

remove redundant quasi-components from a list of regular semi-algebraic systems

 

Calling Sequence

Parameters

Description

Examples

References

Compatibility

Calling Sequence

RemoveRedundantComponents(lrc, R)

RemoveRedundantComponents(lrsas, R)

Parameters

lrc

-

list of regular chains

lrsas

-

list of regular semi-algebraic systems

R

-

polynomial ring

Description

• 

The command RemoveRedundantComponents(lrc, R) returns a list lrc2 of regular chains whose quasi-components are pairwise noninclusive and such that lrc and lrc2 are Lazard decompositions of the same algebraic variety. Consequently, this command removes from lrc2 those quasi-components that are redundant for inclusion.

• 

The command RemoveRedundantComponents(lrsas, R) returns a list res of regular semi-algebraic system whose zero sets are pairwise noninclusive, and such that lrsas and res have the same zero set.

• 

For more details, see Algorithm 35 in the Ph.D. thesis of Yuzhen Xie.

Examples

withRegularChains:withChainTools:withSemiAlgebraicSetTools:

Consider a polynomial ring with two variables

RPolynomialRingy,x

R:=polynomial_ring

(1)

Consider two regular chains in R

rc1Chainyy+1,EmptyR,R

rc1:=regular_chain

(2)

rc2Chainx,y,EmptyR,R

rc2:=regular_chain

(3)

The solutions of one are contained in those of the other. The redundant one will be removed as follows

outRemoveRedundantComponentsrc1,rc2,R

out:=regular_chain

(4)

mapEquations,out,R

y2+y

(5)

The case of semi-algebraic system.

C10<a&comma;0<b&comma;0<c&comma;a<b&plus;c&comma;b<a&plus;c&comma;c<a&plus;b&comma;b2&plus;a2c20&colon;

C20<a&comma;0<b&comma;0<c&comma;a<b&plus;c&comma;b<a&plus;c&comma;c<a&plus;b&comma;cb2&plus;a2c22<ab22acc2&plus;a2b2&colon;

C3ac<0&comma;0<a&comma;0<b&comma;0<c&comma;a<b&plus;c&comma;b<a&plus;c&comma;c<a&plus;b&colon;

SC1&comma;C2&comma;C3

S:=0<a&comma;0<b&comma;0<c&comma;a<b&plus;c&comma;b<a&plus;c&comma;c<a&plus;b&comma;a2&plus;b2c20&comma;0<a&comma;0<b&comma;0<c&comma;a<b&plus;c&comma;b<a&plus;c&comma;c<a&plus;b&comma;ca2&plus;b2c22<ab2a2&plus;2ac&plus;b2c2&comma;ac<0&comma;0<a&comma;0<b&comma;0<c&comma;a<b&plus;c&comma;b<a&plus;c&comma;c<a&plus;b

(6)

RPolynomialRinga&comma;b&comma;c&colon;

dec1mapop&comma;mapRealTriangularize&comma;S&comma;R

dec1:=regular_semi_algebraic_system&comma;regular_semi_algebraic_system&comma;regular_semi_algebraic_system&comma;regular_semi_algebraic_system

(7)

dec2RemoveRedundantComponentsdec1&comma;R

dec2:=regular_semi_algebraic_system

(8)

evalbnopsdec2<nopsdec1

true

(9)

IsContaineddec1&comma;dec2&comma;R

true

(10)

IsContaineddec1&comma;dec2&comma;R

true

(11)

References

  

Xie, Y. "Fast Algorithms, Modular Methods, Parallel Approaches and Software Engineering for Solving Polynomial Systems Symbolically" Ph.D. Thesis, University of Western Ontario, Canada, 2007.

Compatibility

• 

The RegularChains[SemiAlgebraicSetTools][RemoveRedundantComponents] command was introduced in Maple 16.

• 

The lrsas parameter was introduced in Maple 16.

• 

For more information on Maple 16 changes, see Updates in Maple 16.

See Also

ChainTools

EqualSaturatedIdeals

IsContained

IsIncluded

IsInSaturate

PolynomialRing

RegularChains

 


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