Problem
ALG-B3-M07-P011 Self-Shift
#11
★★★★☆ Level 4 of 5
Find all \(P\in\mathbb R[x]\) if \(P(x+P(y))=P(x)+P(y)\).
If the degree is greater than \(1\), compare degree in \(y\).
If \(P\) is constant, then \(P=0\). Let \(d\ge1\). The degree of the left-hand side in \(y\) is \(d^2\), while the right-hand side has degree \(d\), so \(d=1\). Let \(P(x)=ax+b\), \(a\ne0\). Substitution gives \(a(x+ay+b)+b=ax+b+ay+b\). Hence \(a^2=a\), so \(a=1\); any constant works. The answer is \(P=0\) and \(P(x)=x+b\).
Similar to the previous problem, but with a family of shifts.