Let be any (commutative, with identity) ring. A polynomial in the ring is said to be homogeneous of degree if there exists elements (with , , and not all are zero) such that
Homogeneous polynomials are important in algebraic geometry because they are used to build stuff in projective space. That’s possible precisely because homogeneous polynomials are those which behave well under a scaling transformation of their variables:
Proposition. A polynomial is homogeneous of degree if and only if the relation holds between these elements of .
Notice that there’s an induction principle for homogeneous polynomials of a given degree : to prove a proposition is true about them, it suffices to show that it is true for monomials of degree , and that the proposition is stable under sums of homogeneous polynomials of degree . This technique may be used to easily prove the following:
Euler’ Theorem. Let be an homogeneous polynomial of degree . Then
Proof. We proceed by Noetherian induction as indicated above. For the base case, suppose is a monomial , with a multi-index having , and a coefficient. Differentiating with respect to some fixed variable yields zero if , otherwise it yields where is the multi-index obtained from by decreasing its th component by one. In both cases, multiplying the resulting expression by gives the term . Taking the sum of these terms for from to and factoring out the common factor yields the desired equation.
For the induction step, suppose and are two homogeneous polynomials of degree such that the equation holds for them; we want to show it also holds for their sum . This is obvious, since the partial differential is a linear operator.
The following result generalizes Euler’s Theorem (just take ) and can be proven using the same technique:
Proposition. Let be a homogeneous polynomial of degree . For any natural number , we have where the sum is taken over all tuples .