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)




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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


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




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$\#\{\alpha \in \mathbb{C}: G(r(\alpha),s(\alpha))=0\}$ 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.




---


Andrew Granville ask for other $G$. There are constructions.


Here is example in computer readable form with t=x:



```
G=x^3*y + x*y^3 + 8*y^4
r=x^16 - 40*x^14 - 4352*x^13 + 348*x^12 + 1024*x^11 + 189416*x^10 + 14080*x^9 + 622022*x^8 + 4485120*x^7 + 910312*x^6 + 13647104*x^5 + 65163612*x^4 + 3943424*x^3 + 46235608*x^2 + 134216960*x - 1050623
s=16*x^15 + 176*x^13 + 5248*x^12 + 400*x^11 + 30976*x^10 + 433584*x^9 - 4224*x^8 + 343472*x^7 + 486912*x^6 - 392816*x^5 - 4060288*x^4 - 16662352*x^3 + 1313024*x^2 + 8413200*x + 33685632

```


Counterexample to Proposition of Granville related to abc conjecture
I checked the proof of Granville. The proof only yields the bound
$\max \{ \deg(r),\deg(s)\}(\deg(G)-2)\}+1$
which covers your counterexample.


To be more detailed: The polynomials $r,s$ yield a morphism $\mathbb{P}^1\to \mathbb{P}^1$. The set $G=0$ consists of $\deg(G)$ points. The set of $\alpha$ with $G(r(\alpha),s(\alpha)=0$ is the set of points in the preimage of this set with finite $t$-coordinate. The complete set consists of at least 
$\max \{ \deg(r),\deg(s)\}(\deg(G)-2)\}+2$
points, but one of them may be the point at infinity and hence you have to subtract one. This is precisely what is happening in your example.


