MathLabs

Problem 6

Let PP be a non-constant polynomial with integer coefficients. If n(P)n(P) is the number of distinct integers kk such that P(k)2=1P(k)^2=1, prove that n(P)−deg⁡(P)≤2n(P)-\deg(P)\le2.
Step 4 of 4: Finish the bound
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n(P)≤d+2⟹n(P)−deg⁡(P)≤2n(P)\le d+2\quad\Longrightarrow\quad n(P)-\deg(P)\le2
Detailed analysis

The first level equation contributes at most d roots, and the other contributes at most two additional roots. Hence n(P) is at most d+2, which is the required inequality.