Is an algebraic set unique?

Question

Let $\vec x$ be coordinates in $\mathbb R^n$. Let there be an algebraic set, $X$, defined by \begin{align} P_1(\vec x)&=0, \\ . \\ .\\ .\\ P_m(\vec x)&=0, \end{align} where $m<n$. Can $X$ be realised by another set of $m$ polynomial equations not related by overall scaling, or is the set of polynomial equations unique?

Answer 1

Yes, there are cases where $X$ can be realized by other sets of equations. For instance, if all your polynomials are linear, then $X$ is a linear subspace. If you've done some linear algebra, you'll know that there are infinitely many ways to represent a subspace as a solution set of linear equations.