Does anyone know of Invariant Theory enough to comment on this question?

Question

I am trying to find a minimal set of invariants for the binary homogenous form $$\displaystyle ax^7 + bx^{6}y + cx^{5}y^{2} + dx^{4}y^{3} + ex^{3}y^{4} + fx^{2}y^{5} + gxy^{6} + hy^{7}$$ What is the basis for all of the invariants for this form? Is there an easier way like using properties of symmetry without going through the crazy calculation? I've already calculated the binary form when the leading term's degree is 2,3 and 4 using the software SAGE.

Answer 1

• Dixmier, Jacques; Lazard, D. (1988), "Minimum number of fundamental invariants for the binary form of degree 7", Journal of Symbolic Computation 6 (1): 113–115

  Abstract. The minimal number of fundamental invariants for the binary form of degree 7 was a problem left open since last century. It has been solved partly by computer algebra, partly by hand computations.

This looks promising.

Also see "On complete system of invariants for the binary form of degree 7", Leonid Bedratyuk.