View source for Implicit function (in algebraic geometry)

A function given by an algebraic equation. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050320/i0503201.png" /> be a polynomial in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050320/i0503202.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050320/i0503203.png" /> (with complex coefficients, say). Then the variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050320/i0503204.png" /> of zeros of this polynomial can be regarded as the graph of a correspondence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050320/i0503205.png" />. This correspondence, allowing for a certain impreciseness, is also called the function given implicitly by the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050320/i0503206.png" />. Generally speaking, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050320/i0503207.png" /> is many-valued and not defined everywhere and so is not a function in the usual sense. There are two ways of turning this correspondence into a function. The first, which goes back to B. Riemann, consists in assuming that the domain of definition of the implicit function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050320/i0503208.png" /> is not <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050320/i0503209.png" /> but the variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050320/i05032010.png" />, which is a finite-sheeted covering of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050320/i05032011.png" />. This device leads to the highly important concept of a [[Riemann surface|Riemann surface]]. In this approach the notion of an implicit function interlinks with that of an [[Algebraic function|algebraic function]].

The other approach consists in representing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050320/i05032012.png" /> locally as the graph of a single-valued function. Various implicit-function theorems assert that there are open sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050320/i05032013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050320/i05032014.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050320/i05032015.png" /> is the graph of a smooth function (in one sense or another) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050320/i05032016.png" /> (see [[Implicit function|Implicit function]]). However, the open sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050320/i05032017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050320/i05032018.png" /> are, as a rule, not open in the [[Zariski topology|Zariski topology]] and have no meaning in algebraic geometry. Therefore, one modifies this method in the following manner. A formal germ (or branch) at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050320/i05032019.png" /> of the implicit function given by the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050320/i05032020.png" /> is defined as a [[Formal power series|formal power series]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050320/i05032021.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050320/i05032022.png" />. Quite generally, a power series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050320/i05032023.png" /> satisfying a polynomial equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050320/i05032024.png" /> is said to be algebraic. An algebraic power series converges in a certain neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050320/i05032025.png" />.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050320/i05032026.png" /> be a local [[Noetherian ring|Noetherian ring]] with maximal ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050320/i05032027.png" />. An element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050320/i05032028.png" /> of the completion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050320/i05032029.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050320/i05032030.png" /> is said to be algebraic over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050320/i05032031.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050320/i05032032.png" /> for some polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050320/i05032033.png" />. The set of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050320/i05032034.png" /> that are algebraic over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050320/i05032035.png" /> forms a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050320/i05032036.png" />. The following version of the implicit-function theorem shows that there are sufficiently many algebraic functions. Let

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050320/i05032037.png" /></td> </tr></table>

be a collection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050320/i05032038.png" /> polynomials from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050320/i05032039.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050320/i05032040.png" /> be elements of the residue class field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050320/i05032041.png" /> (the bar above a letter means reduction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050320/i05032042.png" />) such that:

1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050320/i05032043.png" />;

2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050320/i05032044.png" />.

Then there exist elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050320/i05032045.png" /> algebraic over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050320/i05032046.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050320/i05032047.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050320/i05032048.png" />. In other words, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050320/i05032049.png" /> is a [[Hensel ring|Hensel ring]].

Another result of this type is Artin's approximation theorem (see [[#References|[2]]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050320/i05032050.png" /> be a local ring that is the localization of an algebra of finite type over a field. Next, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050320/i05032051.png" /> be a system of polynomial equations with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050320/i05032052.png" /> (or in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050320/i05032053.png" />) and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050320/i05032054.png" /> be a vector with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050320/i05032055.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050320/i05032056.png" />. Then there is a vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050320/i05032057.png" /> with components in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050320/i05032058.png" />, arbitrarily close to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050320/i05032059.png" /> and such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050320/i05032060.png" />. There is also a version [[#References|[3]]] of this theorem for systems of analytic equations.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M. Artin,   "Algebraic spaces" , Yale Univ. Press  (1971)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M. Artin,   "Algebraic approximation of structures over complete local rings"  ''Publ. Math. IHES'' , '''36'''  (1969)  pp. 23–58</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M. Artin,   "On the solution of algebraic equations"  ''Invent. Math.'' , '''5'''  (1968)  pp. 277–291</TD></TR></table>