Recently @salim-barzani asked a question about a paper that involved analysing the different types of roots of polynomials. The appendix in that paper gave the example of the roots of using the analysis in Lu et al, "A complete discrimination system for polynomials", Science in China (Ser. E), 39 (1996) 628-646. The analysis uses the discriminant sequence and extensions. Maple provides this through RegularChains:-ParametricSystemTools:-DiscrminantSequence. For example for this polynomial we find there is a real root of multiplicity 2 and a complex conjugate pair when  where the  are the th entries in the discriminant sequence .

The problem with these conditions is that they are in terms of the  and not directly in terms of the  parameters. One can derive these conditions and then solve them to find the conditions on the parameters, but Maple has various routines in the RegularChains, RootFinding:-Parametric and SolveTools packages that directly find conditions on parameters to find when there are specified numbers of real or complex roots for polynomial systems. So this post is my attempt to use these tools to find the conditions on the parameters of the above polynomial that give various types of roots. One immediate difficulty is that generally these routines count distinct roots irrespective of multiplicity, and so some indirect analysis is required. There are several different types of commands and analyses that could be used, and my choices here are more to do with my learning experience than an optimum analysis.

The first conclusion is that it is possible, although RealComprehensiveTriangularize did not work as I expected when asking for zero real roots (bug?). Assuming it had worked, RealComprehensiveTriangularize could cover all the cases here, though that will not be true when there are more parameters. There doesn't seem to be an obvious systematic way of doing this analysis, which is a downside. Another downside is the large number of subcase conditions, which look as if they could be combined into fewer subcases. CellDecomposition works well for cases without multiplicity.