9.75a Stack
The SquareFreeRegularTriangularSet domain constructor implements
square-free regular triangular sets. See the
RegularTriangularSet domain constructor for general regular
triangular sets. Let T be a regular triangular set consisting
of polynomials t1, ..., tm ordered by increasing main variables.
The regular triangular set T is square-free if T is empty
or if t1, ..., tm-1 is square-free and if the polynomial
tm is square-free as a univariate polynomial with coefficients in the
tower of simple extensions associated with t1, ..., tm-1.
The main interest of square-free regular triangular sets is that their
associated towers of simple extensions are product of fields.
Consequently, the saturated ideal of a square-free regular triangular
set is radical. This property simplifies some of the operations
related to regular triangular sets. However, building square-free
regular triangular sets is generally more expensive than building
general regular triangular sets.
As the RegularTriangularSet domain constructor, the
SquareFreeRegularTriangularSet domain constructor also implements a
method for solving polynomial systems by means of regular triangular
sets. This is in fact the same method with some adaptations to take
into account the fact that the computed regular chains are
square-free. Note that it is also possible to pass from a
decomposition into general regular triangular sets to a decomposition
into square-free regular triangular sets. This conversion is used
internally by the LazardSetSolvingPackage package constructor.
N.B. When solving polynomial systems with the
SquareFreeRegularTriangularSet domain constructor or the
LazardSetSolvingPackage package constructor, decompositions have no
redundant components. See also LexTriangularPackage and
ZeroDimensionalSolvePackage for the case of algebraic systems with a
finite number of (complex) solutions.
We shall explain now how to use the constructor
SquareFreeRegularTriangularSet.
This constructor takes four arguments.
The first one, R, is the coefficient ring of the polynomials;
it must belong to the category GcdDomain.
The second one, E, is the exponent monoid of the polynomials;
it must belong to the category OrderedAbelianMonoidSup.
the third one, V, is the ordered set of variables;
it must belong to the category OrderedSet.
The last one is the polynomial ring;
it must belong to the category RecursivePolynomialCategory(R,E,V).
The abbreviation for SquareFreeRegularTriangularSet is
SREGSET.
Note that the way of understanding triangular decompositions
is detailed in the example of the RegularTriangularSet
constructor.
Let us illustrate the use of this constructor with one example
(Donati-Traverso). Define the coefficient ring.
Type: Domain
Define the list of variables,
ls : List Symbol := [x,y,z,t]
Type: List Symbol
and make it an ordered set;
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Type: Domain
then define the exponent monoid.
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Type: Domain
Define the polynomial ring.
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Type: Domain
Let the variables be polynomial.
Type:
NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])
Type:
NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])
Type:
NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])
Type:
NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])
Now call the SquareFreeRegularTriangularSet domain constructor.
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Type: Domain
Define a polynomial system.
p1 := x ** 31 - x ** 6 - x - y
Type:
NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])
Type:
NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])
Type:
NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])
Type:
List NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])
First of all, let us solve this system in the sense of Kalkbrener.
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Type:
List SquareFreeRegularTriangularSet(Integer,
IndexedExponents OrderedVariableList [x,y,z,t],
OrderedVariableList [x,y,z,t],
NewSparseMultivariatePolynomial(Integer,
OrderedVariableList [x,y,z,t]))
And now in the sense of Lazard (or Wu and other authors).
zeroSetSplit(lp,false)$ST
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Type:
List SquareFreeRegularTriangularSet(Integer,
IndexedExponents OrderedVariableList [x,y,z,t],
OrderedVariableList [x,y,z,t],
NewSparseMultivariatePolynomial(Integer,
OrderedVariableList [x,y,z,t]))
Now to see the difference with the RegularTriangularSet domain
constructor, we define:
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Type: Domain
and compute:
lts := zeroSetSplit(lp,false)$T
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Type:
List RegularTriangularSet(Integer,
IndexedExponents OrderedVariableList [x,y,z,t],
OrderedVariableList [x,y,z,t],
NewSparseMultivariatePolynomial(Integer,
OrderedVariableList [x,y,z,t]))
If you look at the second set in both decompositions in the sense of Lazard,
you will see that the polynomial with main variable y is not the same.
Let us understand what has happened.
We define:
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Type:
RegularTriangularSet(Integer,
IndexedExponents OrderedVariableList [x,y,z,t],
OrderedVariableList [x,y,z,t],
NewSparseMultivariatePolynomial(Integer,
OrderedVariableList [x,y,z,t]))
Type:
Union(
NewSparseMultivariatePolynomial(Integer,
OrderedVariableList [x,y,z,t]),...)
tower := collectUnder(ts,'y)$T
Type:
RegularTriangularSet(Integer,
IndexedExponents OrderedVariableList [x,y,z,t],
OrderedVariableList [x,y,z,t],
NewSparseMultivariatePolynomial(Integer,
OrderedVariableList [x,y,z,t]))
pack := RegularTriangularSetGcdPackage(R,E,V,P,T)
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Type: Domain
Then we compute:
toseSquareFreePart(pol,tower)$pack
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Type:
List Record(val:
NewSparseMultivariatePolynomial(Integer,
OrderedVariableList [x,y,z,t]),
tower: RegularTriangularSet(Integer,
IndexedExponents OrderedVariableList [x,y,z,t],
OrderedVariableList [x,y,z,t],
NewSparseMultivariatePolynomial(Integer,
OrderedVariableList [x,y,z,t])))