Proving this function to be identically equal to $0$

Asked Sep 08 '21 at 05:55

Active Sep 08 '21 at 06:00

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This question was asked in a master's exam for which I am preparing and I was unable to solve it. So, I am asking for help here.

Let $f\in \mathbb{R}[x,y]$ be such that there exists a non-empty open set $U\subseteq \mathbb{R}^2$ such that $f(x,y)=0$ for every $(x,y)\in U$. Show that $f=0$.

Th problem is that I am not able to think which result I should use. I have read theory in Multivariable calculus from Apostol's real analysis and understood it but I still solving problems of Multivariable calculus is a weak point for me.

So, Can you please tell me which propositions should be used?

edited Sep 08 '21 at 06:00

Infinity_hunter

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asked Sep 08 '21 at 05:55

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I believe if there are analyticity assumptions then you have your result – FShrike Sep 08 '21 at 05:58
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@FShrike You are given $f\in\mathbb{R}[x,y]$, i.e., $f$ is a polynomial. – user10354138 Sep 08 '21 at 06:10

Proving this function to be identically equal to $0$

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