Rank of a Polynomial function over Finite Fields

Question

Consider a function $f:\mathbb{R}^k \to \mathbb{R}^n$ where $f=(f_1,f_2,\dots,f_n)$ with each $f_i$ being a polynomial function of variables $t_1,t_2,\dots,t_k$.

Now if the polynomials $f_1,f_2,\dots,f_k$ were linear functions of $t_1,\dots,t_k$, then $f$ is a linear map from $\mathbb{R}^k$ to $\mathbb{R}^n$. In this case we can define a rank of the map, and we have the rank-nullity theorem relating the rank and the dimension of its kernel.

What if the functions $f_1,f_2,\dots,f_n$ were of higher degree?

What would be the natural generalization of the "rank" of the map $f$? In one context, I noticed that they defined the rank of $f$ to be the rank of the Jacobian matrix of $f$ (whose entries are treated as elements of the rational function field over $\mathbb{R}$. Is this a standard notion? It does generalize the notion of rank of a linear map.

Do we have some analogous rank-nullity theorem in this case, relating the rank and the dimension of the variety $f=0$? I am aware that there are standard rigorous notions of dimension for varieties.

I am especially interested in such a notion of rank when the underlying field is not $\mathbb{R}$ but a small prime field, say $\mathbb{F}_2$. In this case, the Jacobian seems a bit restrictive, since partial derivatives over $\mathbb{F}_2$ are themselves not very useful. Or maybe I am wrong here. Another notion I have seen is that of algebraic independence, which seems reasonable as well. Would that agree with a rank-nullity theorem? More importantly, would that agree with the rank of the Jacobian?

So my essential question is:

What would be a notion of rank for a polynomial function $f:\mathbb{F}_2^k \to \mathbb{F}_2^n$?

Answer 1

I propose a definition based on the Jacobian idea. There another kind of derivative called the Hasse derivative.

Let $P(\mathbf{X})=\sum_{\beta}c_\beta\mathbf{X}^\beta$ be a polynomial in $k$ indeterminates $\mathbf{X}=(X_1,\ldots,X_k)$. The partial derivative of $P$ with respective to the multi-index $\alpha$ is defined as $\partial^\alpha P(\mathbf{X})=\sum_{\beta\ge\alpha}{\beta\choose\alpha}c_\beta\mathbf{X}^{\beta-\alpha}$.

The Hasse derivative of $P$ with respect to the multi-index $\alpha$ is defined as $\mathcal{D}^\alpha P(\mathbf{X})=\sum_{\beta\ge\alpha}\alpha!{\beta\choose\alpha}c_\beta\mathbf{X}^{\beta-\alpha}=\alpha!\partial^\alpha P(\mathbf{X})$. Note that here ${\beta\choose\alpha}=\prod_{i=1}^k{\beta_i\choose\alpha_i}$ and $\alpha!=\prod_{i=1}^k(\alpha_i!)$.

The Jacobian can then be defined using the Hasse derivatives instead of the partial derivatives.