I want to impose properties on a general function $f(\mathbf{x})$ such that the only function that satisfies those properties is: $$\sum\nolimits_{i=1}^{n}\lambda _{i}x_{i}^{\alpha }.$$ In other words what are the properties that uniquely characterize the function: $$\sum\nolimits_{i=1}^{n}\lambda _{i}x_{i}^{\alpha }.$$ As far as I know this summation is called a diagonal form or an homogeneous polynomial without cross-terms.
Please help me or give me a reference in which such result is obtained.
Many thanks in advance
Hint: (assuming $\alpha \in \mathbb{N}$) consider functions $f : \mathbb{R}^n \to \mathbb{R}$ infinitely differentiable, which satisfy:
$$\frac{\partial^2 f}{\partial x_i \partial x_j} = 0 \quad \;\;\text{for all}\;\; 1 \le i \ne j \le n$$
$$\frac{\partial^{k} f}{\partial x_i^{k}} = 0 \quad \;\;\text{for all}\;\; 1 \le i \le n \;\;\text{and}\;\; k \ge \alpha + 1$$