remove redundant quasi-components from a list of regular chains - Maple Help

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RegularChains[ChainTools][RemoveRedundantComponents] - remove redundant quasi-components from a list of regular chains

RegularChains[SemiAlgebraicSetTools][RemoveRedundantComponents] - remove redundant quasi-components from a list of regular semi-algebraic systems

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

R:=PolynomialRingy,x

R:=polynomial_ring

(1)

Consider two regular chains in R

rc1:=Chainyy+1,EmptyR,R

rc1:=regular_chain

(2)

rc2:=Chainx,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

out:=RemoveRedundantComponentsrc1,rc2,R

out:=regular_chain

(4)

mapEquations,out,R

y2+y

(5)

The case of semi-algebraic system.

C1:=0<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;

C2:=0<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;

C3:=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&colon;

S:=C1&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)

R:=PolynomialRinga&comma;b&comma;c&colon;

dec1:=mapop&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)

dec2:=RemoveRedundantComponentsdec1&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)

See Also

ChainTools, EqualSaturatedIdeals, IsContained, IsIncluded, IsInSaturate, PolynomialRing, RegularChains

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.


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