RegularChains[ParametricSystemTools]
PreComprehensiveTriangularize
compute a pre-comprehensive triangular decomposition
Calling Sequence
Parameters
Description
Examples
PreComprehensiveTriangularize(sys, d, R)
sys
-
list of polynomials
d
number of parameters
R
polynomial ring
The command PreComprehensiveTriangularize(sys, d, R) returns a pre-comprehensive triangular decomposition of sys, with respect to the last d variables of R.
A pre-comprehensive triangular decomposition is a refined triangular decomposition (in the Lazard sense) with additional properties, aiming at studying parametric polynomial systems.
Let U be the last d variables of R, which we regard as parameters. A finite set S of regular chains of R forms a pre-comprehensive triangular decomposition of F with respect to U, if for every parameter value u, there exists a subset Su of S such that
(1) the regular chains of Su specialize well at u, and
(2) after specialization at u, these chains form a triangular decomposition (in the Lazard sense) of the polynomial system F specialized at u. See the command DefiningSet for the term specialize well.
withRegularChains:
withConstructibleSetTools:
withParametricSystemTools:
R≔PolynomialRingx,y,s
R≔polynomial_ring
F≔s−y+1x,s−x+1y
A pre-comprehensive triangular decomposition of F consists of three regular chains.
pctd≔PreComprehensiveTriangularizeF,1,R
pctd≔regular_chain,regular_chain,regular_chain
mapInfo,pctd,R
y+1x−s,y2+y−s,x+1,y+1,s,x,y,s
Compare it with the output of Triangularize.
dec≔TriangularizeF,R,output=lazard
dec≔regular_chain,regular_chain
mapInfo,dec,R
y+1x−s,y2+y−s,x+1,y+1,s
See Also
ComprehensiveTriangularize
ConstructibleSet
DefiningSet
DiscriminantSet
Info
RegularChains
Triangularize
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