Problem 6

Find all polynomials

P

in two variables such that, for a positive integer

n

P(tx,ty)=t^nP(x,y)

for all real

t,x,y

; for all real

a,b,c

P(b+c,a)+P(c+a,b)+P(a+b,c)=0

; and

P(1,0)=1

Step 4 of 6: Show that

Q

depends only on

x+y

Previous step Next step

Q(x+d,y-d)=Q(x,y)\quad\text{for all }d

Detailed analysis

The map $(x,y)\mapsto(2y,x-y)$ preserves $x+y$ and has eigenvalues $1,-2$ . Iterating Step 3 gives infinitely many points $(x+d,y-d)$ on the same line, so the polynomial in $d$ , $Q(x+d,y-d)-Q(x,y)$ , has infinitely many roots and is identically zero. Thus $Q(x+d,y-d)=Q(x,y)$ for every real $d$ .