Implicit function (in algebraic geometry)

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. In this approach the notion of an implicit function interlinks with that of an 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). However, the open sets $ U $ and $ W $ are, as a rule, not open in the 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 $ 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 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.

Another result of this type is Artin's approximation theorem (see [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 [3] of this theorem for systems of analytic equations.

References

[1]	M. Artin, "Algebraic spaces" , Yale Univ. Press (1971) MR0427316 MR0407012 Zbl 0232.14003 Zbl 0226.14001 Zbl 0216.05501
[2]	M. Artin, "Algebraic approximation of structures over complete local rings" Publ. Math. IHES , 36 (1969) pp. 23–58 MR0268188 Zbl 0181.48802
[3]	M. Artin, "On the solution of algebraic equations" Invent. Math. , 5 (1968) pp. 277–291