Dimension polynomial

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.