Let $M \subset \mathbb{R}^n$ be an analytic manifold. Let $p_i$, $i=1,\cdots,m$ be $m$ polynomials on reals defined in $\mathbb{R}^n$ and $m < n$.
We assume that there is a linear dependency of $p_i$ on $M$, i.e. there is a $\beta \in \mathbb{R}^{n-1}$ such that $$ p_n(x) = \sum_{i=1}^{m-1} \beta_i p_i(x) $$ when $x \in M$.
I wonder if the linear dependency restricted to $M$ can be extended to $\mathbb{R}^n$?
Suppose $n \geqslant 2$. Define $M= \mathbb{R}\times \{0\}^{n-1} \subset \mathbb{R}^n$, and $p_1 = X_2$, $p_2 =\left({X_2}\right)^2$, with $p_1,p_2 \in \mathbb{R}[X_1,\ldots,X_n]$. Then $p_1|_{M} = p_2|_{M} = 0$ but $p_1$ and $p_2$ are not colinear. Hence, there is a linear dependency in $M$ that does not extend to the ambiant space.
The same thing with $\mathbb{R}^k \times \{0\}^{n-k} \subset \mathbb{R}^n$, $k \geqslant 1$, and $p_1 = X_{n}$ and $p_2 = (X_n)^2$ shows that the result is false in all positive codimensions.