Editing Zero of a function (section)

=== Applications ===
In [[algebraic geometry]], the first definition of an [[algebraic variety]] is through zero sets. Specifically, an [[affine algebraic set]] is the [[set intersection|intersection]] of the zero sets of several polynomials, in a [[polynomial ring]] <math>k\left[x_1,\ldots,x_n\right]</math> over a [[field (mathematics)|field]]. In this context, a zero set is sometimes called a ''zero locus''.

In [[Mathematical analysis|analysis]] and [[geometry]], any [[closed set|closed subset]] of <math>\mathbb{R}^n</math> is the zero set of a [[smooth function]] defined on all of <math>\mathbb{R}^n</math>. This extends to any [[smooth manifold]] as a corollary of [[paracompactness]]. <!-- There is obvious overlap between this and the next paragraph, but it takes someone more experienced to merge the two. -->

In [[differential geometry]], zero sets are frequently used to define [[manifold]]s. An important special case is the case that <math>f</math> is a [[smooth function]] from <math>\mathbb{R}^p</math> to <math>\mathbb{R}^n</math>. If zero is a [[regular value]] of <math>f</math>, then the zero set of <math>f</math> is a smooth manifold of dimension <math>m=p-n</math> by the [[Submersion_(mathematics)#Local_normal_form|regular value theorem]].

For example, the unit <math>m</math>-[[sphere]] in <math>\mathbb{R}^{m+1}</math> is the zero set of the real-valued function <math>f(x)=\Vert x \Vert^2-1</math>.