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.
.
Rational approach: p, q coprime ⇝ Bézout up + vq = 1.
Look for x = b−u
tq
, y = bv
tp
+ H.O.T., b ∈ K
×
. Indeed,
f0(b−u
tq
, bv
tp
) =
X
pi+qj=r
ai,j b−uj
tqj
bvi
tpi
= tr
X
pi+qj=r
ai,j bv(i0+qk)−u(j0−pk)
= tr
bvi0−uj0
X
pi+qj=r
ai,j bk
= tr
bvi0−uj0
B(b).
Solve B(b) = 0, plug in x = b−u
xq
1 , y = bv
xp
1 (1 + y1), and
iterate until the equation is nonsingular in y.
Nicolas Mascot Algebraic curves