Computing the rational parametrisations
f =
X
pi+qj=r
ai,j xj
yi
| {z }
f0(x,y)
+
X
pi+qj>r
ai,j xj
yi
| {z }
H.O.T.
.
Puiseux approach: Look for roots of valuation p/q, so
y = bxp/q
+ H.O.T. with b ∈ K
×
determined by f0(x, y) = 0:
f0(x, bxp/q
) =
X
pi+qj=r
ai,j xqj/q
bi
xpi/q
= xr/q
X
pi+qj=r
ai,j bi
= xr/q
B(b).
But as p, q coprime, i = i0 + qk, j = j0 − pk for k ∈ Z,
so B(b) is actually a polynomial in bq
⇝ q-th roots ⇝ bad for
rationality.
Nicolas Mascot Algebraic curves