I was reading the following paper: "A Short Proof for the Krull Dimension of a Polynomial Ring. Thierry Coquand and Henri Lombardi" and came across this corollary. (This is present with a better explanation in this paper as well - "Hidden constructions in abstract algebra (3) Krull dimension, going-up, going-down".)
Corollary 3 Let $k$ be a field, and let $R$ be a commutative $k$-algebra. If any sequence $x_0, . . . , x_l$ in $R$ is algebraically dependent over $k$, then the Krull dimension of $R$ is at most $l$.
In the proof they mention this: Let $Q(x_0, . . . , x_l) = 0$ be an algebraic dependence relation over $k$. Then $Q$ can be represented in the following way, $ Q = {x_0}^{m_0}\cdots {x_l}^{m_l} + {x_0}^{m_0}\cdots {x_l}^{m_l +1}R_l + {x_0}^{m_0}\cdots {x_{l-1}}^{m_{l-1} +1}R_{l-1} + \cdots + {x_0}^{m_0}{x_1}^{m_1 + 1} R_1 + {x_0}^{m_0 + 1}R_0 $
My doubt is how can we obtain this expression from an algebraic dependence relation $Q$ ? Consider an example, given the $k$-algebra, $k[x,y,z]/\langle xy + 2, z + 1\rangle$, $\bar{x},\bar{y}$ are algebraically dependent but how can we represent it using this equation? How will the constant $2$ be represented?
