Let $\Omega$ be a non-empty open set of $\mathbb{R}^2$ and $P$ a polynomial of two variables, such that $P(x,y) = 0$ for all $(x,y) \in \Omega$. a) Show that for all $(x,y) \in \Omega$, the open set $\Omega$ contains a subset of the form $I \times J$, where $I$ and $J$ are non-empty open intervals of $\mathbb{R}$ containing $x$ and $y$ respectively. The use of a drawing will be appreciated; however, this drawing will not constitute a proof. b) Deduce that $P$ is the zero polynomial. One may reduce to studying polynomial functions of one variable.
Let $\Omega$ be a non-empty open set of $\mathbb{R}^2$ and $P$ a polynomial of two variables, such that $P(x,y) = 0$ for all $(x,y) \in \Omega$.
a) Show that for all $(x,y) \in \Omega$, the open set $\Omega$ contains a subset of the form $I \times J$, where $I$ and $J$ are non-empty open intervals of $\mathbb{R}$ containing $x$ and $y$ respectively.
The use of a drawing will be appreciated; however, this drawing will not constitute a proof.
b) Deduce that $P$ is the zero polynomial.
One may reduce to studying polynomial functions of one variable.