Rosen's paper (J. B. Rosen. Existence and uniqueness of equilibrium points for concave n-person games. Econometrica, 33(3):520–534, 1965) presents a condition for the uniqueness of the Nash Equilibrium in $n$ players game. The setup is described in simple details by this other economics exchange post. My curiosity is why is the Jacobian of $g(\mathbf{s},\mathbf{z})$, where $g(\mathbf{s},\mathbf{z})$ is the pseudogradient of the function $\sigma$, a square matrix? If $s \in \mathbb{R}^m$ and there are $n$ players, it seems to me that the matrix would be $n \times m$ instead of $m \times m$. I have left out many of the details of the problem set up as they are covered in the other post; however, I can restate the set up or provide additional information if necessary.
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While there are $n$ players, a player's strategy is not single dimensional. The total dimension of the strategies is $m$. So the vector of strategies $s$ is $m$ dimensional. $g(s,z)$ is also $m$ dimensional, which is why the Jacobian is $m \times m$
