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VJIMC 2015 Category I, Problem 4

Source: VJIMC2015

August 10, 2015
polynomialcollege contestsnumber theory

Problem Statement

Problem 4 Let mm be a positive integer and let pp be a prime divisor of mm. Suppose that the complex polynomial a0+a1x+…+anxna_0 + a_1x + \ldots + a_nx^n with n<pp−1φ(m)n < \frac{p}{p-1}\varphi(m) and an≠0a_n \neq 0 is divisible by the cyclotomic polynomial ϕm(x)\phi_m(x). Prove that there are at least pp nonzero coefficients ai .a_i\ .
The cyclotomic polynomial ϕm(x)\phi_m(x) is the monic polynomial whose roots are the mm-th primitive complex roots of unity. Euler’s totient function φ(m)\varphi(m) denotes the number of positive integers less than or equal to mm which are coprime to mm.
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