Yesterday, I wrote a post on the Mason-Stothers theorem and presented an elementary proof of the theorem given by Noah Snyder. As mentioned in that post, I will present now a problem proposed by Magkos Athanasios (Kozani, Greece) that can be solved almost “effortlessly” using the aforesaid theorem.
Problem: Let and be polynomials with complex coefficients and let be a complex number. Prove that if
for all , then the polynomials and are constants.
Solution: First, note that if is a constant, then this forces to be a constant, and vice-versa. Now, suppose and are not constants. We show that this leads to a contradiction.
Observe that if and have a common root, say, , then we have , which implies , which implies , a contradiction. Therefore, we conclude and are relatively prime polynomials, and hence, and are also relatively prime. Now, let and . Then, from the given equation, we conclude and for some .
Now, applying the Mason-Stothers theorem, we get
, which implies , a contradiction! And, we are done.