First, we check the very existence of the partial Legendre transform. Consider the following Jacobian and Hessian matrices:

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$$\frac{\partial(y_{i},y_{p})}{\partial(s,t)}=\left(\begin{array}[]{cc}\frac{\partial^{2}K}{\partial s_{j}\partial s_{i}}&\frac{\partial^{2}K}{\partial t_{q}\partial s_{i}}\\ 0&\mathbbm{1}\end{array}\right)=\left(\begin{array}[]{cc}V&B\\ 0&\mathbbm{1}\end{array}\right),\quad\operatorname{Hess}K(s,t)=\left(\begin{array}[]{cc}V&B\\ {}^{t}B&-W\end{array}\right).$$

Then

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$$\frac{\partial(s,t)}{\partial(y_{i},y_{p})}=\left(\begin{array}[]{cc}V^{-1}&-V^{-1}B\\ 0&\mathbbm{1}\end{array}\right),\quad\operatorname{Hess}\Psi(y)=\left(\begin{array}[]{cc}V^{-1}&-V^{-1}B\\ {}^{t}(-V^{-1}B)&W+{{}^{t}B}V^{-1}B\end{array}\right).$$

This shows that if both $V$ and $W$ are symmetric and positive definite, then $\operatorname{Hess}\Psi$ is also a symmetric positive definite matrix, so there exists (locally) a convex function $\Psi$ – the partial Legendre transform.

On the other hand the inverse of a non-degenerate $2\times 2$ block matrix with $\det A\neq 0$ and $\det D\neq 0$ is:

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$$\left(\begin{array}[]{cc}A&B\\ C&D\end{array}\right)^{-1}=\left(\begin{array}[]{cc}(A-BD^{-1}C)^{-1}&(-A+BD^{-1}C)^{-1}BD^{-1}\\ (-D+CA^{-1}B)^{-1}CA^{-1}&(D-CA^{-1}B)^{-1}\end{array}\right),$$

which implies that

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$$\det\left(\begin{array}[]{cc}A&B\\ C&D\end{array}\right)=1\quad\Longleftrightarrow\quad\det A^{-1}=\det(D-CA^{-1}B).$$

Applied to $\operatorname{Hess}\Psi$ the last observation shows that $\Psi$ is a local Monge-Ampère solution if and only if $\det V=\det W$. ∎