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 $V_l$.

  • $\begingroup$ you can edit your original post to clarify $\endgroup$
    – Zag
    Dec 23, 2022 at 12:24
  • 1
    $\begingroup$ Have you tried $(x+\zeta^ky)^l$ for $k=0,\dots,l-1$ and $\zeta=e^{2\pi i/l}$? $\endgroup$
    – Somos
    Dec 23, 2022 at 14:52
  • $\begingroup$ @zag thank you. $\endgroup$
    – user536450
    Dec 23, 2022 at 16:25
  • $\begingroup$ @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 ? $\endgroup$
    – user536450
    Dec 23, 2022 at 16:30
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    $\begingroup$ Please show your work for $l=2$ in your question. Why would you need complex square roots? $\endgroup$
    – Somos
    Dec 23, 2022 at 18:42


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