# A simple basis for the space of two variables homogenous polynomials of a fixed degree.

Denote by $$\mathcal{P}_l$$ the linear space of the complex-valued homogeneous polynomials on $$\mathbb{R}^2$$ of degree $$l$$. I read that polynomials of the form $$(ax+by)^l$$ where $$a,b\in \mathbb{C}$$ do span the space $$\mathcal{P}_l$$. Can anyone help why this is true? Many thanks in advance.\

In the book I read this he gives an indication as follows : if we denote by $$V_l$$ the span of the $$(ax+by)^l$$ where $$a,b\in \mathbb{C}$$. Remark that since $$V_l$$ is a subspace of $$\mathcal{P_l}\equiv \mathbb{C}^{l+1}$$ it is a closed subspace and thus if $$\gamma(t)$$ is a smooth curve in $$V_l$$, the derivative $$\gamma'(t)$$ will also lie in $$V_l$$.

• you can edit your original post to clarify
– Zag
Dec 23, 2022 at 12:24
• Have you tried $(x+\zeta^ky)^l$ for $k=0,\dots,l-1$ and $\zeta=e^{2\pi i/l}$? Dec 23, 2022 at 14:52
• @zag thank you. Dec 23, 2022 at 16:25
• @Somos. Thank you. Can I have a little bit more detail and where does the idea of the l-th root of unity come from ? Dec 23, 2022 at 16:30
• Please show your work for $l=2$ in your question. Why would you need complex square roots? Dec 23, 2022 at 18:42