RegularChains[ParametricSystemTools] - Maple Programming Help

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RegularChains[ParametricSystemTools]

 ComplexRootClassification
 compute a classification of the complex roots of a polynomial system depending on parameters

 Calling Sequence ComplexRootClassification(F, d, R) ComplexRootClassification(F, H, d, R) ComplexRootClassification(CS, d, R)

Parameters

 F - list of polynomials H - list of polynomials d - number of parameters R - polynomial ring CS - constructible set

Description

 • The integer d must be positive and smaller than the number of variables.
 • The characteristic of R must be zero and the last d variables of R are regarded as parameters.
 • For a parametric algebraic system, this command computes all the possible numbers of solutions of this system together with the corresponding necessary and sufficient conditions on its parameters.
 • More precisely, let V be the variety defined by F. The command ComplexRootClassification(F, d, R) returns a classification of the complex roots of F depending on parameters, that is, a finite partition P of the parameter space into constructible sets such that above each part, the number of solutions of V is either infinite or constant.
 • If a constructible set CS is specified, the representing regular systems of CS must be square-free. The function call ComplexRootClassification(CS, d, R) returns a classification of the points of the constructible set CS, that is, a finite partition P of the parameter space into constructible sets such that above each part, the number of solutions of CS is either infinite or constant.
 • If H is specified, let $W$ be the variety defined by the product of polynomials in H. The command ComplexRootClassification(F, H, d, R) returns a classification of the points of the constructible set V-W depending on parameters.

Examples

 > $\mathrm{with}\left(\mathrm{RegularChains}\right):$
 > $\mathrm{with}\left(\mathrm{ConstructibleSetTools}\right):$
 > $\mathrm{with}\left(\mathrm{ParametricSystemTools}\right):$
 > $R≔\mathrm{PolynomialRing}\left(\left[x,y,s\right]\right)$
 ${R}{:=}{\mathrm{polynomial_ring}}$ (1)
 > $F≔\left[s-\left(y+1\right)x,s-\left(x+1\right)y\right]$
 ${F}{:=}\left[{s}{-}\left({y}{+}{1}\right){}{x}{,}{s}{-}\left({x}{+}{1}\right){}{y}\right]$ (2)

The computation below shows that the input parametric system can have 1 solution or 2 distinct solutions. The corresponding conditions on the parameters are given by constructible sets.

 > $\mathrm{CC}≔\mathrm{ComplexRootClassification}\left(F,1,R\right)$
 ${\mathrm{CC}}{:=}\left[\left[{\mathrm{constructible_set}}{,}{1}\right]{,}\left[{\mathrm{constructible_set}}{,}{2}\right]\right]$ (3)

These constructible sets are printed below.

 > $\mathrm{map}\left(x→\left[\mathrm{Info}\left({x}_{1},R\right),{x}_{2}\right],\mathrm{CC}\right)$
 $\left[\left[\left[\left[{4}{}{s}{+}{1}\right]{,}\left[{1}\right]\right]{,}{1}\right]{,}\left[\left[\left[{}\right]{,}\left[{s}{,}{4}{}{s}{+}{1}\right]\right]{,}\left[\left[{s}\right]{,}\left[{1}\right]\right]{,}{2}\right]\right]$ (4)