Let $\rm{SL}_n$ be the special linear group and let $\rm{Sym}_n$ be the set of all symmetric matrices of size n. $\rm{SL}_n$ acts on $(\rm{Sym}_n)^m$ by $g(A_1, \ldots , A_m)=(gA_1 g^{\rm T}, \ldots , g A_m g^{\rm T})$. Clearly, in the case of $m=1$ the ring of invariants is generated by $\det(A)$. But what are the invariants of this group action in general? Is there an easy description of the ring of invariants, e.g. by giving the generators?

If you are willing to replace $SL$ by $GL$ (so without the determinant) then this is a special case of a much more general result. Apply the theory in the following paper to the quiver with one vertex and $m$ edges. MR0958897 (90e:16048) Le Bruyn, Lieven; Procesi, Claudio Semisimple representations of quivers. Trans. Amer. Math. Soc. 317 (1990), no. 2, 585–598. The special case you are considering was studied before this. The reference has already been given by Agol. 


For $n=2$ an answer is classically known, see Grace and Young, p.161, 139A. Probably there is a modern explanation of this result known, too. 


This is part of classical invariant theory: the study of joint invariants of several quadratic forms. As far as I know these rings of invariants are only known in a few special cases. There has been work by Turnbull and Todd for the $n=3$ case. A recent paper on the subject which contains pointers to the classical literature is ``La théorie des invariants des formes quadratiques ternaires revisitée'' by Bruno Blind. 

