I want to prove that any three relatively prime polynomials $A, B, C \in \mathbb C[X]$ verifying $A+B+C=0$ have at least $1+\max(\deg A, \deg B, \deg C)$ distinct roots in total among each other.

I have tried using the gcd to characterize the number of distinct roots, as for a polynomial $P$ its number of distinct roots is given by the degree $P$ minus that of $\gcd(P,P')$. As $A, B, C$ are relatively prime and $A+B+C=0$ I can show that $A$ and $B$ are relatively prime and assume without loss of generality that $\deg A=\deg B=n$ and $\deg A+B \leq n$. So I want to show that $ABC=-(A^2B+AB^2)$ has at least $n+1$ distinct roots.

I tried considering the rank of the Sylvester matrices $S_A, S_B$ and $S_{A+B}=S_A+S_B$ of $(A,A')$, $(B,B')$ and $(A+B,A'+B')$ respectively, but this seems a dead end (inequalities for the rank of the sum being too wide, and it does not exploit that $A,B$ are relatively prime).

I have also considered reasoning by induction as the property is clearly true for $n=1$, but again this does not seem to yield much... Any ideas?



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