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

RegularChains[SemiAlgebraicSetTools][PartialCylindricalAlgebraicDecomposition] - compute a partial cylindrical algebraic decomposition

Calling Sequence

PartialCylindricalAlgebraicDecomposition(p, lp, R)

Parameters

R

-

polynomial ring

p

-

polynomial of R

lp

-

list of polynomials of R

Description

• 

The command PartialCylindricalAlgebraicDecomposition returns llr a list of points in the Euclidean space of dimension d, where d the number of variables in R.

• 

Each point in llr is a sample point of a d dimensional connected open set, which is a cell of a Cylindrical Algebraic Decomposition (CAD) induced by the polynomial p and the polynomials in lp, under the variable projection order given by R. Recall that the variables in R are sorted in decreasing order.

• 

If lp is not an empty list, then the points which do not satisfy q > 0 for all polynomial q in lp are discarded; otherwise, the points are in one-to-one correspondence to all the d dimensional CAD cells.

• 

The coordinates of all these points are rational numbers, and the ith coordinate of each point of llr corresponds the ith variable of R.

• 

The base field of R is the field of rational numbers.

Examples

withRegularChains:

withSemiAlgebraicSetTools:

R:=PolynomialRingx,y,t:

PartialCylindricalAlgebraicDecompositiony,,R

0,12,0,0,12,0

(1)

PartialCylindricalAlgebraicDecompositiony,x,R

12,12,0,12,12,0

(2)

PartialCylindricalAlgebraicDecompositionyx,,R

12,12,0,12,12,0,12,12,0,12,12,0

(3)

PartialCylindricalAlgebraicDecompositionyx,x,R

12,12,0,12,12,0

(4)

See Also

CylindricalAlgebraicDecompose, RealRootClassification, RealTriangularize , RegularChains, SamplePoints, Triangularize 


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