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diff --git a/src/hyper/pages/WUTSET.ht b/src/hyper/pages/WUTSET.ht new file mode 100644 index 00000000..d41587a8 --- /dev/null +++ b/src/hyper/pages/WUTSET.ht @@ -0,0 +1,128 @@ +% 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}{9.87} +% +% ===================================================================== +\begin{page}{WuWenTsunTriangularSetXmpPage}{9.87 WuWenTsunTriangularSet} +% ===================================================================== +\beginscroll +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. +\endscroll +\autobuttons +\end{page} +% |