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If a polynomial vanishes on a substantial portion of an irreducible variety then it vanishes...

caffeinemachine

Well-known member
MHB Math Scholar
Mar 10, 2012
823
India
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Definitions.

Definition.
A real affine algebraic variety is a subset $V$ of $\R^n$ which occurs as the common zero set of a collection of polynomials in $\R[x_1 , \ldots, x_n]$. We say that $V$ is irreducible if $V$ cannot be written as a union of two proper subvarieties.

Definition.
Let $V$ be an affine algebraic variety in $\R^n$. Let $I(V)$ denote the set of all the polynomials in $n$ variables which vanish on each point of $V$. Note that the set $\set{Df_p:\ f\in I(V)}$ is a real vector space, and is a subspace of $(\R^n)^*$. (Here $Df_p$ denotes the derivative of $f$ at the point $p$). It is therefore finite dimensional. The rank of $V$ at $p$, written $\rank_p(V)$, is defined as the dimension of this vector space. The rank of $V$ is defined as
\begin{equation}
\rank(V) = \max_{p\in V}\rank_p(V)
\end{equation}
A point $p\in V$ is said to be a smooth point if $\rank_p(V)=\rank(V)$. A smooth point may also be referred to as a simple point or a non-singular point. The set of all the simple points of $V$ will be denoted by $\reg(V)$. A point in $V$ which is not simple is called singular. The set of all the singular points of $V$ will be denoted by $\sing(V)$.


Questions

Question.
Let $V$ be an irreducible affine algebraic variety in $\R^n$ and $g$ be a polynomial in $\R[x_1 , \ldots, x_n]$ which vanishes in a Euclidean neighborhood of a simple point of $V$. Then it is necessarily true that $g$ vanishes on the whole of $V$?