local parameter on an irreducible affine algebraic curve On page 14 of Shafarevich's Basic Algebraic Geometry 1, it is stated that for an irreducible affine algebraic curve $X: f(x,y) = 0$, and a nonsingular point $P \in X$, there is a regular function $t$ (called the local parameter) that vanishes at $P$, and such that for every rational function $u$ that is not identically $0$ on $X$, $u = t^k v$ for some $k$ and some regular function $v$ where $v(P) \neq 0$.
Now, it is proved in the case where $P = (0,0)$ and where $\dfrac{\partial f}{\partial y} \neq 0$, and in this case we get $t = x$.
He calls $k$ the multiplicity of the zero of $P$ on $u$.
My questions are:
(1) How would our function $t$ change if we didn't have $P = (0,0)$ or $\dfrac{\partial f}{\partial y} \neq 0$?
(2) What is the significance of the number $k$? In particular, is it related at all to the multiplicity of a point on $X$ ($m_P(X)$ as in Fulton's Algebraic Curves)?
(3) In the absence of any commutative algebra machinery (e.g. local rings), how can I make sense of any of this?!
For the answers, it would be nice if the use of things like local rings, etc..., could be kept to a minimum. I am more or less familiar with just pages 1-14 of Shafarevich.
EDIT: I realize this is asking for a lot, so any partial answers will get votes, and in time, I'll select the best one.
 A: Some quick answers:


*

*We can always perform an affine-linear change of coordinates to make the assumptions $P=(0,0)$ and $\partial f / \partial y \neq 0$ true in the new coordinates. (Note that $P$ is a nonsingular point of the curve, at least one of $\partial f / \partial x$ and  $\partial f / \partial y$ must be nonzero there.) If you want to translate back to the original coordinates, the function $t$ will then have the form $t=ax+by+c$ for some constants $a,b,c$. 

*No, the multiplicity of a rational function is not related to the multiplicity of a point. (Indeed, we are talking about a nonsingular point $P$ on $X$, so $P$ always has multiplicity one.) As Shafarevich's terminology indicates, you are supposed to think of this as the order of vanishing of the function $u$ at the point $p$. It might help to recall the case of functions on $\mathbf C$: a meromorphic function $f$ can be expanded in a Laurent series around 0 of the form
$$\sum_{k=N}^\infty a_k z^k $$
with $a_N \neq 0$; the standard definition is that $f$ vanishes to order $N$ at $0$. (Note that $N$ might be negative!) To see the similarity with your definition, rewrite the sum as
$$z^N \left(\sum_{k=0}^\infty a_{k+N} z^K \right)$$
and note that the term in parentheses in nonzero at $0$. 

*I hope that my answer to 2. helps with this. 
