|
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 with rational coefficients, representing equations of the form =0
|
ineqs
|
-
|
list of polynomials with rational coefficients, representing constraint inequalities of the form
|
|
|
|
|
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
|
|
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 , and solutions at . 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 (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
|
|
>
|
|
>
|
|
The discriminant variety in this first example is .
The case gives a solution of the first equation at . In the case , the second and third equations coincide and therefore the system becomes underdetermined and has infinitely many solutions. Finally, the case corresponds to a boundary case for the inequality .
In the following univariate example, compute the well-known discriminant of a quadratic polynomial.
>
|
|
The next example illustrates the alternate calling sequence.
>
|
|
| (3) |
The next system has a solution only if ; it is not generically solvable.
>
|
|
This system has solutions of multiplicity greater than for all parameter values.
>
|
|
This system has infinitely many solutions for all parameter values.
>
|
|
The following example represents 2 lines and 2 points in space. It has more equations than indeterminates.
>
|
|
|
|
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.
|
|
|
|