MathLabs

Problem 6

Find all polynomials PP in two variables such that, for a positive integer nn, P(tx,ty)=tnP(x,y)P(tx,ty)=t^nP(x,y) for all real t,x,yt,x,y; for all real a,b,ca,b,c, P(b+c,a)+P(c+a,b)+P(a+b,c)=0P(b+c,a)+P(c+a,b)+P(a+b,c)=0; and P(1,0)=1P(1,0)=1.
Step 1 of 6: Use homogeneity
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In plain words

The scaling law says every monomial has total degree nn.

P(x,y)=∑k=0nckxkyn−kP(x,y)=\sum_{k=0}^{n}c_kx^ky^{n-k}
Detailed analysis

The first condition is exactly homogeneity of degree nn.