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''of an extension of differential fields''
 
''of an extension of differential fields''
  
 
A polynomial describing the number of derivative constants in the solution of a system of partial differential equations; it is an analogue of the [[Hilbert polynomial|Hilbert polynomial]].
 
A polynomial describing the number of derivative constants in the solution of a system of partial differential equations; it is an analogue of the [[Hilbert polynomial|Hilbert polynomial]].
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032490/d0324901.png" /> be a differential [[Extension of a differential field|extension of a differential field]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032490/d0324902.png" />. A maximal subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032490/d0324903.png" /> that is differentially separably independent over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032490/d0324904.png" /> is called a differential inseparability basis. A differential inseparability basis of an extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032490/d0324905.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032490/d0324906.png" /> that is differentially algebraically independent over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032490/d0324907.png" /> is called a differential transcendence basis.
+
Let $  G $
 
+
be a differential [[Extension of a differential field|extension of a differential field]] $  F $ .  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032490/d0324908.png" /> be a finitely-generated differential extension, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032490/d0324909.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032490/d03249010.png" /> be a generic zero of a prime differential ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032490/d03249011.png" />. The differential transcendence degree of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032490/d03249012.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032490/d03249013.png" /> is called the differential dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032490/d03249014.png" /> (it is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032490/d03249015.png" />). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032490/d03249016.png" /> is another prime differential ideal with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032490/d03249017.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032490/d03249018.png" /> and, moreover, equality can occur even for a strict inclusion. For this reason, it is desirable to have a finer measure for measuring ideals.
+
A maximal subset of $  G $
 
+
that is differentially separably independent over $  F $
A filtration of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032490/d03249019.png" /> of differential polynomials by the degrees of the derivations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032490/d03249020.png" /> induces a filtration of the extension fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032490/d03249021.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032490/d03249022.png" />:
+
is called a differential inseparability basis. A differential inseparability basis of an extension $  G $
 
+
over $  F $
<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/d/d032/d032490/d03249023.png" /></td> </tr></table>
+
that is differentially algebraically independent over $  F $
 
+
is called a differential transcendence basis.
There exists (see [[#References|[2]]]) a polynomial whose value at points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032490/d03249024.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032490/d03249025.png" /> is equal to the transcendence degree of the extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032490/d03249026.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032490/d03249027.png" />. It is called the dimension polynomial of the extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032490/d03249028.png" />, and has the form
 
  
<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/d/d032/d032490/d03249029.png" /></td> </tr></table>
+
Let  $  G $
 +
be a finitely-generated differential extension,  $  G = F \langle  \eta _{1} \dots \eta _{n} \rangle $ ,
 +
and let  $  ( \eta _{1} \dots \eta _{n} ) $
 +
be a generic zero of a prime differential ideal  $  p \subset F \{ Y _{1} \dots Y _{n} \} $ .  
 +
The differential transcendence degree of  $  G $
 +
over  $  F $
 +
is called the differential dimension of  $  p $ (
 +
it is denoted by  $  d (p) $ ).  
 +
If  $  q $
 +
is another prime differential ideal with  $  p \subset q $ ,
 +
then  $  d (p) \geq d (q) $
 +
and, moreover, equality can occur even for a strict inclusion. For this reason, it is desirable to have a finer measure for measuring ideals.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032490/d03249030.png" /> is the cardinality of the set of derivation operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032490/d03249031.png" /> and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032490/d03249032.png" /> are integers. The dimension polynomial is a birational invariant of the field, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032490/d03249033.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032490/d03249034.png" />, but it is not a differential birational invariant, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032490/d03249035.png" /> does not in general imply that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032490/d03249036.png" />. Nevertheless, this polynomial includes differential birational invariants, such as the degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032490/d03249037.png" /> of the polynomial (called the differential type of the extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032490/d03249038.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032490/d03249039.png" />), and the leading coefficient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032490/d03249040.png" /> (called the typical differential dimension). Among the differential dimension polynomials corresponding to various systems of differential generators of a differential extension there exists a minimal one with respect to some order relation on the set of all numerical polynomials, which is hence a differential birational invariant of the extension.
+
A filtration of the ring  $  F \{ Y _{1} \dots Y _{n} \} $
 +
of differential polynomials by the degrees of the derivations  $  \theta \in \Theta $
 +
induces a filtration of the extension fields  $  G = F \langle  \eta _{1} \dots \eta _{n} \rangle $
 +
of  $  F $ :
 +
$$
 +
=
 +
{\mathcal G} _{0}  \subset  {\mathcal G} _{1}  \subset \dots .
 +
$$
 +
There exists (see [[#References|[2]]]) a polynomial whose value at points  $  s \in \mathbf Z $
 +
for all  $  s \geq N _{0} $
 +
is equal to the transcendence degree of the extension  $  {\mathcal G} _{s} $
 +
of  $  F $ .  
 +
It is called the dimension polynomial of the extension  $  G /F $ ,
 +
and has the form $$
 +
\omega _ {\eta /F} (x)  = 
 +
\sum _ {0 \leq i \leq m}
 +
a _{i}
 +
\binom{x + i}{i} ,
 +
$$
 +
where  $  m $
 +
is the cardinality of the set of derivation operators $  \Delta $
 +
and the $  a _{i} $
 +
are integers. The dimension polynomial is a birational invariant of the field, that is, $  F ( \eta ) = F ( \zeta ) $
 +
implies  $  \omega _ {\eta /F} = \omega _ {\zeta /F} $ ,  
 +
but it is not a differential birational invariant, that is, $  F \langle  \eta \rangle = F \langle  \zeta \rangle $
 +
does not in general imply that $  \omega _ {\eta /F} = \omega _ {\zeta /F} $ .  
 +
Nevertheless, this polynomial includes differential birational invariants, such as the degree $  \tau = \mathop{\rm deg}\nolimits \  \omega _ {\eta /F} $
 +
of the polynomial (called the differential type of the extension $  F \langle  \eta \rangle $
 +
over $  F \  $ ),  
 +
and the leading coefficient $  a _ \tau  $ (
 +
called the typical differential dimension). Among the differential dimension polynomials corresponding to various systems of differential generators of a differential extension there exists a minimal one with respect to some order relation on the set of all numerical polynomials, which is hence a differential birational invariant of the extension.
  
 
The differential dimension polynomial is also defined for differential modules.
 
The differential dimension polynomial is also defined for differential modules.

Latest revision as of 20:47, 17 December 2019


of an extension of differential fields

A polynomial describing the number of derivative constants in the solution of a system of partial differential equations; it is an analogue of the Hilbert polynomial.

Let $ G $ be a differential extension of a differential field $ F $ . A maximal subset of $ G $ that is differentially separably independent over $ F $ is called a differential inseparability basis. A differential inseparability basis of an extension $ G $ over $ F $ that is differentially algebraically independent over $ F $ is called a differential transcendence basis.

Let $ G $ be a finitely-generated differential extension, $ G = F \langle \eta _{1} \dots \eta _{n} \rangle $ , and let $ ( \eta _{1} \dots \eta _{n} ) $ be a generic zero of a prime differential ideal $ p \subset F \{ Y _{1} \dots Y _{n} \} $ . The differential transcendence degree of $ G $ over $ F $ is called the differential dimension of $ p $ ( it is denoted by $ d (p) $ ). If $ q $ is another prime differential ideal with $ p \subset q $ , then $ d (p) \geq d (q) $ and, moreover, equality can occur even for a strict inclusion. For this reason, it is desirable to have a finer measure for measuring ideals.

A filtration of the ring $ F \{ Y _{1} \dots Y _{n} \} $ of differential polynomials by the degrees of the derivations $ \theta \in \Theta $ induces a filtration of the extension fields $ G = F \langle \eta _{1} \dots \eta _{n} \rangle $ of $ F $ : $$ F = {\mathcal G} _{0} \subset {\mathcal G} _{1} \subset \dots . $$ There exists (see [2]) a polynomial whose value at points $ s \in \mathbf Z $ for all $ s \geq N _{0} $ is equal to the transcendence degree of the extension $ {\mathcal G} _{s} $ of $ F $ . It is called the dimension polynomial of the extension $ G /F $ , and has the form $$ \omega _ {\eta /F} (x) = \sum _ {0 \leq i \leq m} a _{i} \binom{x + i}{i} , $$ where $ m $ is the cardinality of the set of derivation operators $ \Delta $ and the $ a _{i} $ are integers. The dimension polynomial is a birational invariant of the field, that is, $ F ( \eta ) = F ( \zeta ) $ implies $ \omega _ {\eta /F} = \omega _ {\zeta /F} $ , but it is not a differential birational invariant, that is, $ F \langle \eta \rangle = F \langle \zeta \rangle $ does not in general imply that $ \omega _ {\eta /F} = \omega _ {\zeta /F} $ . Nevertheless, this polynomial includes differential birational invariants, such as the degree $ \tau = \mathop{\rm deg}\nolimits \ \omega _ {\eta /F} $ of the polynomial (called the differential type of the extension $ F \langle \eta \rangle $ over $ F \ $ ), and the leading coefficient $ a _ \tau $ ( called the typical differential dimension). Among the differential dimension polynomials corresponding to various systems of differential generators of a differential extension there exists a minimal one with respect to some order relation on the set of all numerical polynomials, which is hence a differential birational invariant of the extension.

The differential dimension polynomial is also defined for differential modules.

The dimension polynomial has been computed for the extensions given by the following systems (see [1], where the dimension polynomial is called the measure of rigidity of the system of equations of a field): 1) the wave equation; 2) the Maxwell equation for empty space; 3) the equation of the electromagnetic field given by potentials; and 4) the Einstein equation for empty space. Other examples of the computation of dimension polynomials can be found in [3].

References

[1] A. Einstein, "The meaning of relativity" , Princeton Univ. Press (1953) MR0058336 MR0053649 Zbl 0050.21208
[2] E.R. Kolchin, "Differential algebra and algebraic groups" , Acad. Press (1973) MR0568864 Zbl 0264.12102
[3] A.V. Mikhalev, E.V. Pankrat'ev, "The differential dimension polynomial of a system of differential equations" , Algebra , Moscow (1980) pp. 57–67 (In Russian) Zbl 0722.12004


Comments

For the notions of differential separable independence, differential transcendence degree and differential algebraic independence cf. Extension of a differential field.

How to Cite This Entry:
Dimension polynomial. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Dimension_polynomial&oldid=44300
This article was adapted from an original article by E.V. Pankrat'ev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article