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? 



Invariants of group action: SL_n acts simultaneously on m symmetric matrices
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.


