View source for Implicit function (in algebraic geometry)

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A function given by an algebraic equation. Let  $  F ( X _ {1} \dots X _ {n} , Y ) $
be a polynomial in  $  X _ {1} \dots X _ {n} $
and  $  Y $(
with complex coefficients, say). Then the variety  $  V ( F  ) \subset  \mathbf C  ^ {n+} 1 $
of zeros of this polynomial can be regarded as the graph of a correspondence  $  y :  \mathbf C  ^ {n} \rightarrow \mathbf C $. 
This correspondence, allowing for a certain impreciseness, is also called the function given implicitly by the equation  $  F ( x , y ) = 0 $. 
Generally speaking,  $  y $
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  $  y $
is not  $  \mathbf C  ^ {n} $
but the variety  $  V ( F  ) $, 
which is a finite-sheeted covering of  $  \mathbf C  ^ {n} $. 
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  $  V ( F  ) $
locally as the graph of a single-valued function. Various implicit-function theorems assert that there are open sets  $  U \subset  \mathbf C  ^ {n} $
and  $  W \subset  \mathbf C $
for which  $  ( U \times W ) \cap V ( F  ) $
is the graph of a smooth function (in one sense or another)  $  y :  U \rightarrow W $(
see [[Implicit function|Implicit function]]). However, the open sets  $  U $
and  $  W $
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  $  a \in \mathbf C  ^ {n} $
of the implicit function given by the equation  $  F ( X , Y ) = 0 $
is defined as a [[Formal power series|formal power series]]  $  y \in \mathbf C [ [ X _ {1} - a _ {1} \dots X _ {n} - a _ {n} ] ] $
such that  $  F ( X , y ) = 0 $. 
Quite generally, a power series  $  y $
satisfying a polynomial equation  $  F ( X , Y ) = 0 $
is said to be algebraic. An algebraic power series converges in a certain neighbourhood of  $  a $.

Let  $  A $
be a local [[Noetherian ring|Noetherian ring]] with maximal ideal  $  \mathfrak m $. 
An element  $  y $
of the completion  $  \widehat{A}  $
of  $  A $
is said to be algebraic over  $  A $
if  $  F ( y) = 0 $
for some polynomial  $  F ( Y) \in A [ Y ] $. 
The set of elements of  $  \widehat{A}  $
that are algebraic over  $  A $
forms a ring  $  \widetilde{A}  $. 
The following version of the implicit-function theorem shows that there are sufficiently many algebraic functions. Let

$$ 
f ( Y)  =  ( f _ {1} ( Y _ {1} \dots Y _ {m} ) \dots f _ {m} ( Y _ {1} \dots Y _ {m} ) )
$$

be a collection of  $  m $
polynomials from  $  A [ Y _ {1} \dots Y _ {m} ] $
and let  $  \overline{y}\; {}  ^ {0} = ( \overline{y}\; _ {0} \dots \overline{y}\; _ {m} ) $
be elements of the residue class field  $  A / \mathfrak m $(
the bar above a letter means reduction  $   \mathop{\rm mod}  \mathfrak m $) 
such that:

1)  $  \overline{f}\; ( \overline{y}\; {}  ^ {0} ) = 0 $;

2)  $  { \mathop{\rm det} ( {\partial  f _ {i} } / {\partial  Y _ {i} } ) } bar ( \overline{y}\; {}  ^ {0} ) \neq 0 $.

Then there exist elements  $  y = ( y _ {1} \dots y _ {m} ) $
algebraic over  $  A $
such that  $  f ( y) = 0 $
and  $  \overline{y}\; = \overline{y}\; {}  ^ {0} $. 
In other words,  $  \widetilde{A}  $
is a [[Hensel ring|Hensel ring]].

Another result of this type is Artin's approximation theorem (see [[#References|[2]]]). Let  $  A $
be a local ring that is the localization of an algebra of finite type over a field. Next, let  $  f ( Y) = 0 $
be a system of polynomial equations with coefficients in  $  A $(
or in  $  \widetilde{A}  $) 
and let  $  \widehat{y}  $
be a vector with coefficients in  $  \widehat{A}  $
such that  $  f ( \widehat{y}  ) = 0 $. 
Then there is a vector  $  \widetilde{y}  $
with components in  $  \widetilde{A}  $, 
arbitrarily close to  $  \widehat{y}  $
and such that  $  f ( \widetilde{y}  ) = 0 $. 
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) {{MR|0427316}} {{MR|0407012}} {{ZBL|0232.14003}} {{ZBL|0226.14001}} {{ZBL|0216.05501}} </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 {{MR|0268188}} {{ZBL|0181.48802}} </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>