MathDB
Grand Finale of SMMC

Source: 2021 Simon Marais, B4

November 2, 2021
polynomialderivativecalculus

Problem Statement

The following problem is open in the sense that the answer to part (b) is not currently known. A proof of part (a) will be awarded 5 points. Up to 7 additional points may be awarded for progress on part (b).
Let p(x)p(x) be a polynomial of degree dd with coefficients belonging to the set of rational numbers Q\mathbb{Q}. Suppose that, for each 1≤k≤d−11 \le k \le d-1, p(x)p(x) and its kkth derivative p(k)(x)p^{(k)}(x) have a common root in Q\mathbb{Q}; that is, there exists rk∈Qr_k \in \mathbb{Q} such that p(rk)=p(k)(rk)=0p(r_k) = p^{(k)}(r_k) = 0. (a) Prove that if dd is prime then there exist constants a,b,c∈Qa, b, c \in \mathbb{Q} such that p(x)=c(ax+b)d. p(x) = c(ax + b)^d. (b) For which integers d≥2d \ge 2 does the conclusion of part (a) hold?
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