I'm considering a 2n-dimensional finite symplectic vector space $(V,\omega)$ over $\mathbb{F}_p$ for $p>2$ prime. In my work, I recently came across functions that are of the form \begin{equation} \varphi:V\rightarrow\mathbb{F}_p\text{ such that }\omega(v,w)=0\Rightarrow\varphi(v+w)=\varphi(v)+\varphi(w), \end{equation} i.e. they are linear when restricted to totally isotropic subspaces.
Such functions form a vector space and I'm currently trying to figure out the dimension of this space.
Since linear functions are included it shoud be greater or equal to $2n$. By choosing some symplecitc basis $e_i,f_i$ with $i=1,...,n$ and writing a vector $v\in V$ as \begin{equation} v=\sum\limits_{i}(v_ie_i+\bar{v}_if_i)=\sum\limits_{i:\,\bar{v}_i\neq 0}\bar{v}_i\bigl(f_i+\frac{v_i}{\bar{v}_i}e_i\bigr)+\sum\limits_{i:\,\bar{v}_i= 0}v_ie_i\mapsto\sum\limits_{i:\,\bar{v}_i\neq 0}\bar{v}_i\varphi\bigl(f_i+\frac{v_i}{\bar{v}_i}e_i\bigr)+\sum\limits_{i:\,\bar{v}_i= 0}v_i\varphi(e_i) \end{equation} it should be smaller or equal to $np+n$.
I'd be very happy about any help with deciding if such functions are always linear and, if they are not (which would make them interesting), where I can find out more about them!