Actually, I am reading something about control theory, but it contains lots of mathematical descriptions that I can't understand. The contents are the following:
Consider the system $\mathcal{D} / k$ where $D=k\langle w \rangle$ is generated by a finite set $w=\left(w_1, \ldots, w_q\right)$. The $w_i$ 's are related by a finite set, $\Xi\left(w, \dot{w}, \ldots, w^{(v)}\right)=0$, of algebraic differential equations. Define the algebraic variety $S$ corresponding to $\Xi\left(\xi^0, \ldots, \xi^\nu\right)=0$ in the $(\nu+1) q$-dimensional affine space with coordinates $$ \xi^j=\left(\xi_1^j, \ldots, \xi_q^j\right), \quad j=0,1, \ldots, \nu $$ Theorem 3 If the system $\mathcal{D} / k$ is flat, the affine algebraic variety $S$ contains at each regular point a straight line parallel to the $\xi^\nu$-axes.
The above condition is not sufficient. Consider the system $\mathcal{D} / \mathbb{R}$ generated by $\left(x_1, x_2, x_3\right)$ satisfying $\dot{x}_1=\left(\dot{x}_2\right)^2+\left(\dot{x}_3\right)^3$. This system does not satisfy the necessary condition: it is not flat. The same system $\mathcal{D}$ can be defined via the quantities $\left(x_1, x_2, x_3, x_4\right)$ related by $\dot{x}_1=\left(x_4\right)^2+\left(\dot{x}_3\right)^3$ and $x_4=\dot{x}_2$. Those new equations now satisfy our necessary criterion.
The so-called "system $\mathcal{D} / k$" is a differential field extension. I have learned something about abstract algebra and have a basic understanding of this.
But I don't know what's affine space, what's affine algebraic variety S, or what's the regular point. I how to verify the necessary condition in Theorem 3 for the example given by the author. Can anybody use the examples above to explain these concepts and how to verify the necessary condition in Theorem 3.