Implicit function (in algebraic geometry)
A function given by an algebraic equation. Let be a polynomial in
and
(with complex coefficients, say). Then the variety
of zeros of this polynomial can be regarded as the graph of a correspondence
. This correspondence, allowing for a certain impreciseness, is also called the function given implicitly by the equation
. Generally speaking,
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
is not
but the variety
, which is a finite-sheeted covering of
. 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 locally as the graph of a single-valued function. Various implicit-function theorems assert that there are open sets
and
for which
is the graph of a smooth function (in one sense or another)
(see Implicit function). However, the open sets
and
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
of the implicit function given by the equation
is defined as a formal power series
such that
. Quite generally, a power series
satisfying a polynomial equation
is said to be algebraic. An algebraic power series converges in a certain neighbourhood of
.
Let be a local Noetherian ring with maximal ideal
. An element
of the completion
of
is said to be algebraic over
if
for some polynomial
. The set of elements of
that are algebraic over
forms a ring
. The following version of the implicit-function theorem shows that there are sufficiently many algebraic functions. Let
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be a collection of polynomials from
and let
be elements of the residue class field
(the bar above a letter means reduction
) such that:
1) ;
2) .
Then there exist elements algebraic over
such that
and
. In other words,
is a Hensel ring.
Another result of this type is Artin's approximation theorem (see [2]). Let be a local ring that is the localization of an algebra of finite type over a field. Next, let
be a system of polynomial equations with coefficients in
(or in
) and let
be a vector with coefficients in
such that
. Then there is a vector
with components in
, arbitrarily close to
and such that
. 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 |
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