If $Q$ is the sum of squares quadratic form $\sum_1^n x_i^2$ over some lattice, then $r_Q(m)$, the number of representations of an integer $m$ by $Q$ (order/sign matter) is sometimes given in a nice formula, as in the case with Jacobi's formula in the case of $n=4$. Can we say something moderate about how $r_Q$ grows? It seems like it shouldn't grow faster than polynomially, but I am struggling to see why in a rigorous way. Since $\sum_1^n x_i^2=m$ is the equation of a sphere (or I suppose an ellipsoid if we transform our lattice to $Z^n$ under a change of variable?), we should be able to bound such solutions by some function of the surface area since there should be some small upper bound for the proportion of lattice points over the surface area and the surface area is a polynomial function of the radius.
I would like to make the argument a bit more rigorous, any advice?
Edit: To be clear, I'm interested in how $r_Q(m)$ grows with $m$.