RootFinding[Parametric] - Maple Programming Help

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RootFinding[Parametric]

 DiscriminantVariety
 compute the discriminant variety of a parametric polynomial system

 Calling Sequence DiscriminantVariety(sys, vars, pars) DiscriminantVariety(eqs, ineqs, vars, pars)

Parameters

 sys - list of equations and strict inequalities between polynomials with rational coefficients vars - list of names; the indeterminates pars - (optional) list of names; the parameters eqs - list of polynomials $f$ with rational coefficients, representing equations of the form $f$=0 ineqs - list of polynomials $g$ with rational coefficients, representing constraint inequalities of the form $0

Description

 • The DiscriminantVariety(sys,vars,pars) calling sequence computes a discriminant variety of the system sys of equations and inequalities with respect to the indeterminates vars and the parameters pars.
 • The DiscriminantVariety(eqs,ineqs,vars,pars) calling sequence computes a discriminant variety of the system

${\left[f=0,0

 of equations and inequalities with respect to the indeterminates vars and the parameters pars.
 • The notion of discriminant variety is a generalization of the discriminant of a univariate polynomial, describing all the critical points of the system, including singularities, solutions of multiplicity greater than $1$, and solutions at $\mathrm{\infty }$. It is a subset of the parameter space of lower dimension. See the article by D. Lazard and F. Rouillier in the References section below for details.
 • A discriminant variety has the following property: It divides the parameter space into open, full-dimensional cells such that the number of solutions of the system sys is constant for parameter values chosen from the same open cell. See CellDecomposition.
 • The input system must satisfy the following properties:
 – There are at least as many equations as indeterminates.
 – At least one and at most finitely many complex solutions exist for almost all complex parameter values (that is, the system is generically solvable and generically zero-dimensional).
 – For almost all complex parameter values, there are no solutions of multiplicity greater than $1$ (that is, the system is generically radical). In particular, the input equations are square-free.
 An error occurs if one of these conditions is violated.
 • The result is returned as a list of lists of polynomials in pars such that the discriminant variety is the union of the set of solutions of the polynomials in each inner list.
 • If pars is not specified, it defaults to all the names in sys that are not indeterminates.
 • This command will attempt to find a minimal discriminant variety, but it may return a proper superset in the case that it does not succeed.
 • The discriminant variety is computed using Groebner basis techniques.
 • This command is part of the RootFinding[Parametric] package, so it can be used in the form DiscriminantVariety(..) only after executing the command with(RootFinding[Parametric]). However, it can always be accessed through the long form of the command by using RootFinding[Parametric][DiscriminantVariety](..).

Examples

 > $\mathrm{with}\left(\mathrm{RootFinding}[\mathrm{Parametric}]\right):$
 > $\mathrm{DiscriminantVariety}\left(\left[a{x}^{2}=1,y+bz=0,y+cz=0,0
 $\left[\left[{a}\right]{,}\left[{c}\right]{,}\left[{b}{-}{c}\right]\right]$ (1)

The discriminant variety in this first example is $\left\{\left(a,b,c\right):a=0\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\mathbf{or}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}b=c\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\mathbf{or}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}c=0\right\}$.

The case $a=0$ gives a solution of the first equation at $\mathrm{\infty }$. In the case $b=c$, the second and third equations coincide and therefore the system becomes underdetermined and has infinitely many solutions. Finally, the case $c=0$ corresponds to a boundary case for the inequality $0.

In the following univariate example, compute the well-known discriminant of a quadratic polynomial.

 > $\mathrm{DiscriminantVariety}\left(\left[{x}^{2}+ax+b=0\right],\left[x\right]\right)$
 $\left[\left[{{a}}^{{2}}{-}{4}{}{b}\right]\right]$ (2)

The next example illustrates the alternate calling sequence.

 > $\mathrm{DiscriminantVariety}\left(\left[{x}^{6}+a{x}^{2}+b\right],\left[a\right],\left[x\right],\left[a,b\right]\right)$
 $\left[\left[{a}\right]{,}\left[{b}\right]{,}\left[{4}{}{{a}}^{{3}}{+}{27}{}{{b}}^{{2}}\right]\right]$ (3)

The next system has a solution only if $a=b$; it is not generically solvable.

 > $\mathrm{DiscriminantVariety}\left(\left[{x}^{6}+a{y}^{2}-a=0,{x}^{6}+a{y}^{2}-b=0\right],\left[x,y\right]\right)$

This system has solutions of multiplicity greater than $1$ for all parameter values.

 > $\mathrm{DiscriminantVariety}\left(\left[{x}^{2}+{y}^{2}={a}^{2},x=a\right],\left[x,y\right]\right)$

This system has infinitely many solutions for all parameter values.

 > $\mathrm{DiscriminantVariety}\left(\left[x=ay,y=az,x={a}^{2}z\right],\left[x,y,z\right]\right)$

The following example represents 2 lines and 2 points in space. It has more equations than indeterminates.

 > $\mathrm{DiscriminantVariety}\left(\left[-{z}^{2}+zx-{y}^{2}+y=0,-{z}^{2}+zx+yx-y-x+1=0,{z}^{2}-zx+{y}^{2}-2y+yx-x+1=0\right],\left[y,z\right]\right)$
 $\left[\left[{x}\right]{,}\left[{x}{-}{1}\right]\right]$ (4)

References

 Lazard, D., and Rouillier, F. "Solving parametric polynomial systems." Journal of Symbolic Computation, Vol. 42 No. 6 (2007): 636 - 667.
 Liang, S., Gerhard, J., Jeffrey, D. J., and Moroz G., "A Package for Solving Parametric Polynomial Systems." ACM Communications in Computer Algebra, Vol. 43 No. 3 (2009): 61 - 72.
 Moroz, G.  "Sur la décomposition réelle et algébrique des systèmes dépendant de paramètres." Ph.D. thesis, l'Universite de Pierre et Marie Curie, Paris, France. 2008.