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I know similar questions have been asked many times but somehow I'm still confused.

Please refer to this question and the comments discussion on the first answer for context.

Let $P(x_1,\cdots,x_r)$ be a homogeneous polynomial such that $P(\alpha_1,\cdots,\alpha_r) = 0$ for an uncountably infinite number of $(\alpha_1,\cdots,\alpha_r) \in \mathbb{R}^r_{+}$ (We can also assume there exists an open ball such that $P(.)$ is zero for all $\alpha$ in that ball). Does that mean $P(.)$ is the zero polynomial?

In my understanding a non-zero homogeneous polynomial in finite number of variables will have a finite number of zeros, hence the result follows. Please correct me if I'm wrong.

Canine360
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    The parenthetical comment is crucial. Otherwise consider $P(x,y)=x-y$. This is homogeneous and vanishes on the uncountable set ${(a,a)|a\in\mathbb R }$, but there is no open ball in $\mathbb R^2$ on which $P$ vanishes. – mweiss Jun 08 '18 at 01:26
  • Thank you so much. I have tried to use this reasoning (though there are some mistakes) in my attempt at this question. I would request you to consider having a look, in case you are interested. Thanks again. https://math.stackexchange.com/questions/2811434/system-of-inequalities-in-multivariate-polynomials – Canine360 Jun 08 '18 at 01:47
  • How does the three-variable polynomial $X^2+Y^2-Z^2$ fit in with your understanding? It has infinitely many zeros, doesn’t it? – Lubin Jun 08 '18 at 04:13

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