Problem
ALG-B3-M08-P010 Polynomial Involution
#10
★★★☆☆ Level 3 of 5
Find all \(P\in\mathbb R[x]\) such that \(P(P(x))=x\).
Compare degrees and reduce to the linear case.
If \(P\) is constant, impossible. If \(\deg P=d\), then \(d^2=1\), so \(d=1\). Let \(P(x)=ax+b\). Then \(a^2x+b(a+1)=x\), hence \(a^2=1\), \(b(a+1)=0\). The answer is \(P(x)=x\) or \(P(x)=-x+b\).
Repeats iteration in a polynomial context.