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 $f:\mathbb{F}_2^k \to \mathbb{F}_2^n$?


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.