Past exam of the mathematics course of the University of Cambridge / 2006 / ii / Paper 1 / 24H / b / Solution

Differentiate the homogeneity identity with respect to $t$ at one to obtain Euler's identityOn $X_a$, its right side is $ma\ne 0$, so $d{p}_x\ne 0$. Thus $a$ is a regular value. The preimage theorem says that the nonempty inverse image of a regular value of a smooth map $M^k\to N^l$ is a smooth submanifold of codimension $l$. Applied to $p:{\mathbb{R}}^k\to \mathbb{R}$, it gives $\dim X_a=k-1$.