% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
\newcommand{\WuWenTsunTriangularSetXmpTitle}{WuWenTsunTriangularSet}
\newcommand{\WuWenTsunTriangularSetXmpNumber}{9.91}
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\begin{page}{WuWenTsunTriangularSetXmpPage}{9.91 WuWenTsunTriangularSet}
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\beginscroll

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The \spadtype{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, {\bf R}, is the coefficient ring of the polynomials;
it must belong to the category \spadtype{IntegralDomain}.
The second one, {\bf E}, is the exponent monoid of the polynomials;
it must belong to the category \spadtype{OrderedAbelianMonoidSup}.
The third one, {\bf V}, is the ordered set of variables;
it must belong to the category \spadtype{OrderedSet}.
The last one is the polynomial ring;
it must belong to the category \spadtype{RecursivePolynomialCategory(R,E,V)}.
The abbreviation for \spadtype{WuWenTsunTriangularSet} is
\spadtype{WUTSET}.

Let us illustrate the facilities by an example.

\xtc{
Define the coefficient ring.
}{
\spadpaste{R := Integer \bound{R}}
}
\xtc{
Define the list of variables,
}{
\spadpaste{ls : List Symbol := [x,y,z,t] \bound{ls}}
}
\xtc{
and make it an ordered set;
}{
\spadpaste{V := OVAR(ls) \free{ls} \bound{V}}
}
\xtc{
then define the exponent monoid.
}{
\spadpaste{E := IndexedExponents V \free{V} \bound{E}}
}
\xtc{
Define the polynomial ring.
}{
\spadpaste{P := NSMP(R, V) \free{R} \free{V} \bound{P}}
}
\xtc{
Let the variables be polynomial.
}{
\spadpaste{x: P := 'x \free{P} \bound{x}}
}
\xtc{
}{
\spadpaste{y: P := 'y \free{P} \bound{y}}
}
\xtc{
}{
\spadpaste{z: P := 'z \free{P} \bound{z}}
}
\xtc{
}{
\spadpaste{t: P := 't \free{P} \bound{t}}
}
\xtc{
Now call the \spadtype{WuWenTsunTriangularSet} domain constructor.
}{
\spadpaste{T := WUTSET(R,E,V,P) \free{R} \free{E} \free{V} \free{P} \bound{T} }
}
\xtc{
Define a polynomial system.
}{
\spadpaste{p1 := x ^ 31 - x ^ 6 - x - y \free{x} \free{y} \bound{p1}}
}
\xtc{
}{
\spadpaste{p2 := x ^ 8  - z \free{x} \free{z} \bound{p2}}
}
\xtc{
}{
\spadpaste{p3 := x ^ 10 - t \free{x} \free{t} \bound{p3}}
}
\xtc{
}{
\spadpaste{lp := [p1, p2, p3] \free{p1} \free{p2} \free{p3} \bound{lp}}
}
\xtc{
Compute a characteristic set of the system.
}{
\spadpaste{characteristicSet(lp)$T \free{lp} \free{T}}
}
\xtc{
Solve the system.
}{
\spadpaste{zeroSetSplit(lp)$T \free{lp} \free{T}}
}


The \spadtype{RegularTriangularSet} and \spadtype{SquareFreeRegularTriangularSet} domain constructors,
and the  \spadtype{LazardSetSolvingPackage}, \spadtype{SquareFreeRegularTriangularSet}
and \spadtype{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 \spadtype{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 \spadtype{RegularTriangularSet}
constructor.
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