In the following univariate example, obtain the well-known discriminant of a quadratic polynomial.
Thus, there are full-dimensional open cells, which you can illustrate graphically:
The number of solutions for on each of these cells can be computed using the following command:
So, the univariate equation has exactly two solutions in the yellow and pink cells, below the parabola, and no solution in the blue cell, above the parabola. In fact, it has one solution of multiplicity exactly on the parabola, but this is a lower-dimensional sub-manifold, which is not considered by this command.
The next two examples illustrate the alternate calling sequences. Since the output of the assignment is the same for m2 and m3, they will each produce the same image using the CellPlot command.
Sample points lying entirely in a negative octant are omitted. For example, the open cells represented by the sample points and from m4:-SamplePoints are not included in m3:-SamplePoints.
The following system has no solutions for almost all parameters values.
In this example, we use two different strategies to compute the sample points.
When we use the strategy witnesspoints, the record m7 contains no ProjectionPolynomials.
m6 and m7 can be visualized with CellPlot.
When EnclosingBox has been applied to the solutions, the enclosing boxes are also displayed by CellPlot.
The option output=witnesspoints usually returns fewer points.
The option method=RC computes the equivalent solution record as the default GB method, but usually gives different sample points.
The following examples illustrate invalid inputs:
This system has solutions of multiplicity greater than for all parameter values.
This system has infinitely many solutions for all parameter values.