Looks like there is counterexample to Proposition related to
abc conjecture. Confusion is likely.


From [RATIONAL AND INTEGRAL POINTS ON QUADRATIC TWISTS OF A GIVEN HYPERELLIPTIC CURVE, Andrew Granville](http://www.dms.umontreal.ca/~andrew/PDF/hyperelliptics.pdf)

---

p. 11, Proposition 2 b

Suppose that $G(x,y) \in \mathbb{C}[x,y]$ is homogeneous
without any repeated factors. For any coprime polynomials
$r(t),s(t) \in \mathbb{C}[t]$, we have

$$
\#\{\alpha \in \mathbb{C}: G(r(\alpha),s(\alpha))=0\} 
\ge \max\{\deg(r),\deg(s)\}(\deg(G)-2) + 2.
$$

---

$\#\{\alpha \in \mathbb{C}: G(r(\alpha),s(\alpha))=0\}$ counts
the distinct zeros and equals the degree of the radical of
$G(r(t),s(t))$.

Explicit counterexample.


Let $G(x,y)=x^4+xy^3,r(t)=8t^3 + 64,s(t)=t^4 - 64t$


We have:

$$
G(r(t),s(t))=\left(8\right) \cdot (t + 2) \cdot (t^{2} - 2 t + 4) \cdot (t^{2} + 4 t - 8)^{2} \cdot (t^{4} - 4 t^{3} + 24 t^{2} + 32 t + 64)^{2}
$$

So $G(r(t),s(t))$ have $9$ distinct zeros.

By the Proposition $9 \ge (( (4\cdot(4-2)+2)=10)$ which is false.


> Q1 Is this really counterexample?

The Proposition is unconditional and this doesn't appear to
contradict abc.

The errata of the paper doesn't address this.