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If you want to distinguish between the first five cases invariants will not do, you need covariants. These cases are examples of so called coincident root loci, you can lookup the papers by my collaborator J. Chipalkatti for the equations for such things. See also Polynomial with two repeated roots

If you want to distinguish between the first five cases invariants will not do, you need covariants. These cases are examples of so called coincident root loci, you can lookup the papers by my collaborator J. Chipalkatti for the equations for such things. See also Polynomial with two repeated roots

Case $2$ is characterized by the vanishing of the Hessian of $F$, and the inequality $F\neq 0$. You can find equations for case $3$ in http://www.springerlink.com/content/wm3318667411003m/

(to which you have to add the inequality that the Hessian is not zero).

For case $4$ you can find defining equations in this paper.

Case $2$ is characterized by the vanishing of the Hessian of $F$, and the inequality $F\neq 0$. You can find equations for case $3$ in my article with Chipalkatti: "The bipartite Brill-Gordan locus and angular momentum". Transform. Groups 11 (2006), no. 3, 341--370. A preprint version is http://arxiv.org/abs/math.AG/0502542

(you have to add the inequality that the Hessian is not zero).

For case $4$ you can find defining equations in this other article with Chipalkatti: "Brill-Gordan Loci, transvectants and an analogue of the Foulkes conjecture". Adv. Math. 208 (2007), no. 2, 491--520. A preprint version is http://arxiv.org/abs/math.AG/0411110

For case $4$ you can find defining equations in this paper.

For case $4$ you can find defining equations in this paper.