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## 9.75 SquareFreeRegularTriangularSet

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.

R := Integer

Integer

Type: Domain

Define the list of variables,

ls : List Symbol := [x,y,z,t]

[x,y,z,t]

Type: List Symbol

and make it an ordered set;

V := OVAR(ls)

OrderedVariableList[x,y,z,t]

Type: Domain

then define the exponent monoid.

E := IndexedExponents V

IndexedExponentsOrderedVariableList[x,y,z,t]

Type: Domain

Define the polynomial ring.

P := NSMP(R, V)

NewSparseMultivariatePolynomial(Integer,OrderedVariableList[x,y,z,t])

Type: Domain

Let the variables be polynomial.

x: P := 'x

x

Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])
y: P := 'y

y

Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])
z: P := 'z

z

Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])
t: P := 't

t

Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])

Now call the SquareFreeRegularTriangularSet domain constructor.

ST := SREGSET(R,E,V,P)

SquareFreeRegularTriangularSet(Integer, IndexedExponentsOrderedVariableList[x,y,z,t], OrderedVariableList[x,y,z,t], NewSparseMultivariatePolynomial(Integer, OrderedVariableList[x,y,z,t]))

Type: Domain

Define a polynomial system.

p1 := x ** 31 - x ** 6 - x - y

x 31-x 6-x-y

Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])
p2 := x ** 8 - z

x 8-z

Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])
p3 := x ** 10 - t

x 10-t

Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])
lp := [p1, p2, p3]

[x 31-x 6-x-y,x 8-z,x 10-t]

Type: List NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])

First of all, let us solve this system in the sense of Kalkbrener.

zeroSetSplit(lp)$ST

[{z 5-t 4,t z y 2+2 z 3 y-t 8+2 t 5+t 3-t 2,(t 4-t)x-t y-z 2}]

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

[{z 5-t 4,t z y 2+2 z 3 y-t 8+2 t 5+t 3-t 2,(t 4-t)x-t y-z 2},{t 3-1,z 5-t,t y+z 2,z x 2-t},{t,z,y,x}]

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:

T := REGSET(R,E,V,P)

RegularTriangularSet(Integer,IndexedExponentsOrderedVariableList[x,y,z,t],OrderedVariableList[x,y,z,t],NewSparseMultivariatePolynomial(Integer,OrderedVariableList[x,y,z,t]))

Type: Domain

and compute:

lts := zeroSetSplit(lp,false)$T

[{z 5-t 4,t z y 2+2 z 3 y-t 8+2 t 5+t 3-t 2,(t 4-t)x-t y-z 2},{t 3-1,z 5-t,t z y 2+2 z 3 y+1,z x 2-t},{t,z,y,x}]

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:

ts := lts.2

{t 3-1,z 5-t,t z y 2+2 z 3 y+1,z x 2-t}

Type: RegularTriangularSet(Integer, IndexedExponents OrderedVariableList [x,y,z,t], OrderedVariableList [x,y,z,t], NewSparseMultivariatePolynomial(Integer, OrderedVariableList [x,y,z,t]))
pol := select(ts,'y)$T

t z y 2+2 z 3 y+1

Type: Union( NewSparseMultivariatePolynomial(Integer, OrderedVariableList [x,y,z,t]),...)
tower := collectUnder(ts,'y)$T

{t 3-1,z 5-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)

RegularTriangularSetGcdPackage(Integer, IndexedExponentsOrderedVariableList[x,y,z,t], OrderedVariableList[x,y,z,t], NewSparseMultivariatePolynomial(Integer, OrderedVariableList[x,y,z,t]), RegularTriangularSet(Integer, IndexedExponentsOrderedVariableList[x,y,z,t], OrderedVariableList[x,y,z,t], NewSparseMultivariatePolynomial(Integer, OrderedVariableList[x,y,z,t])))

Type: Domain

Then we compute:

toseSquareFreePart(pol,tower)$pack

[[val=t y+z 2,tower={t 3-1,z 5-t}]]

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])))
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