Section9.87

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## 9.87 WuWenTsunTriangularSet

The WuWenTsunTriangularSet domain constructor implements the characteristic set method of Wu Wen Tsun. This algorithm computes a list of triangular sets from a list of polynomials such that the algebraic variety defined by the given list of polynomials decomposes into the union of the regular-zero sets of the computed triangular sets. The constructor takes four arguments. The first one, R, is the coefficient ring of the polynomials; it must belong to the category IntegralDomain. 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 WuWenTsunTriangularSet is WUTSET.

Let us illustrate the facilities by an example.

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 WuWenTsunTriangularSet domain constructor.

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

WuWenTsunTriangularSet(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])

Compute a characteristic set of the system.

characteristicSet(lp)$T

{z 5-t 4, t 4 z 2 y 2+2 t 3 z 4 y+(-t 7+2 t 4-t)z 6+t 6 z,(t 3-1)z 3 x-z 3 y-t 3}

Type: Union( WuWenTsunTriangularSet(Integer, IndexedExponents OrderedVariableList [x,y,z,t], OrderedVariableList [x,y,z,t], NewSparseMultivariatePolynomial(Integer, OrderedVariableList [x,y,z,t])),...)

Solve the system.

zeroSetSplit(lp)$T

[{t, z, y, x},{t 3-1, z 5-t 4, z 3 y+t 3, z x 2-t},{z 5-t 4, t 4 z 2 y 2+2 t 3 z 4 y+(-t 7+2 t 4-t)z 6+t 6 z,(t 3-1)z 3 x-z 3 y-t 3}]

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

The RegularTriangularSet and SquareFreeRegularTriangularSet domain constructors, the LazardSetSolvingPackage package constructors as well as, SquareFreeRegularTriangularSet and ZeroDimensionalSolvePackage package constructors also provide operations to compute triangular decompositions of algebraic varieties. These five constructor use a special kind of characteristic sets, called regular triangular sets. These special characteristic sets have better properties than the general ones. Regular triangular sets and their related concepts are presented in the paper ``On the Theories of Triangular sets'' By P. Aubry, D. Lazard and M. Moreno Maza (to appear in the Journal of Symbolic Computation). The decomposition algorithm (due to the third author) available in the four above constructors provide generally better timings than the characteristic set method. In fact, the WUTSET constructor remains interesting for the purpose of manipulating characteristic sets whereas the other constructors are more convenient for solving polynomial systems.

Note that the way of understanding triangular decompositions is detailed in the example of the RegularTriangularSet constructor.

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