International Mathematics Competition
for University Students
2023

IMC2023: Day 1, Problem 3

Problem 3. Find all polynomials \(\displaystyle P\) in two variables with real coefficients satisfying the identity

\(\displaystyle P(x,y)P(z,t)=P(xz-yt,xt+yz). \)

Giorgi Arabidze, Free University of Tbilisi, Georgia

Hint: The polynomials \(\displaystyle (x+iy)^n\) and \(\displaystyle (x-iy)^m\) are trivial complex solutions. Suppose that \(\displaystyle P(x,y)=(x+iy)^n(x-iy)^mQ(x,y)\), where \(\displaystyle Q(x,y)\) is divisible neither by \(\displaystyle x+iy\) nor \(\displaystyle x=iy\) and consider \(\displaystyle Q(x,y)\).

  Solution