Local parametrisations
For each point P = (xP, yP) of C, local parametrisations
x = X(t), y = Y (t)
where X, Y are nonconstant formal power series such that
f X(t), Y (t)


= 0 and X(0) = xP, Y (0) = yP.
We assume X and Y are not both series in tn
for any n ⩾ 2.
Uniqueness: Hope that Parametrisations at P ↔ Points of e
C
above P. But can rescale t ← t′
= ct + O(t2
), c ̸= 0 . . .
Existence: OK if P is nonsingular: can Newton w.r.t. x or y.
But what if P is singular?
Nicolas Mascot Algebraic curves