Manuel Kauers Hagenberg, November 30, 2002 RISC-Linz

Algorithmic Algebraic Geometry

Solution 5

Let F ∈ k[x_{1}, …, x_{n}] be a homogenous polynomial of degree d. We have to show Euler’s equation

d · F = ∑_{i=1}^{n} \frac{∂}{∂x_{i}} F.

Proof. It suffices to prove the claim for one single monomial, since multiplication by the constant d is distributive and the sum on the righthand side is commutative and associative, i.e. can be split into several sums, one per monomial.

So, let

m = ∏_{i=1}^{n} x_{i}^{e_{i}}

be a monomial of F. Since F is homogenous, we know that

∑_{i=1}^{n} e_{i} = deg m = deg F = d.  (1)

Furthermore, we have

x_{j} \frac{∂}{∂x_{j}} m = x_{j} \frac{∂}{∂x_{j}} ∏_{i=1}^{n} x_{i}^{e_{i}} = x_{j} · e_{j} · x_{j}^{e_{j}−1} ∏_{{i=0 \atop i≠j}}^{n} x_{i}^{e_{i}} = e_{j} ∏_{i=1}^{n} x_{i}^{e_{i}} = e_{j} · m (2)

and hence

∑_{j=1}^{n} x_{j} \frac{∂}{∂x_{j}} m =^{(2)} ∑_{j=1}^{n}(e_{j} · m) = m · ∑_{j=1}^{n} e_{j} =^{(1)} m · d.

This completes the proof.