Some applications of model theory to differential Galois theory

Transcription

Some applications of model theory to differential
Galois theory
Javier Moreno
Institut Camille Jordan
Université Claude Bernard Lyon 1
Model Theory in Kirishima
March 4th, 2010
Javier Moreno (UCJ-Lyon 1)
Mod. Th. & Dif. Gal. Th.
Kirishima, March 4th, 2010
1 / 25
Differential Galois Theory
The goal of Differential Galois Theory is to study differential equations
by algebraic means. We would like to have methods, mimicking the
ones from classical Galois theory, to evaluate if the integral of a given
function (or the solution of a differential equation) can be expressed in
terms of “elementary functions”.
2 / 25
Differential Algebra
Definition
Let R be an arbitrary (commutative) ring. A derivation on R is an
additive function ∂ : R → R such that ∂(xy ) = x∂(y ) + y ∂(x) for any
x, y ∈ R. A ring (field) R equipped with a derivation ∂ is what we call a
differential ring (field). By the constants of R, denoted CR , we mean
the set of elements in R where ∂ vanish.
Remark
Notions of ideals (prime, maximal), localizations, &c. can be defined
and developed.
3 / 25
Differential Algebra
Definition
Let R be an arbitrary (commutative) ring. A derivation on R is an
additive function ∂ : R → R such that ∂(xy ) = x∂(y ) + y ∂(x) for any
x, y ∈ R. A ring (field) R equipped with a derivation ∂ is what we call a
differential ring (field). By the constants of R, denoted CR , we mean
the set of elements in R where ∂ vanish.
Remark
Notions of ideals (prime, maximal), localizations, &c. can be defined
and developed.
3 / 25
Classic Galois Theory (Ch. 0)
Let us recall the way (algebraic) Galois theory works. Take, for
convenience, a polynomial P(X ) ∈ F [X ] of degree n with no repeated
roots.
The idea is to associate to this polynomial a certain (finite) group and
extract properties about the solution of the polynomial from the
structural properties of the group.
4 / 25
Let us recall the way (algebraic) Galois theory works. Take, for
convenience, a polynomial P(X ) ∈ F [X ] of degree n with no repeated
roots.
The idea is to associate to this polynomial a certain (finite) group and
extract properties about the solution of the polynomial from the
structural properties of the group.
4 / 25
Let
1
],
i6=j (Xi − Xj )
S = F [X1 , . . . , Xn , Q
and let I be the ideal generated by the P(Xi ) in S. Let M > I
maximal in S.
Define the splitting field of P over F as K = S/M. And let
G = Aut(K /F ). We call this the Galois group of P.
Q
Since S contains i6=j (Xi − Xj ) then the images of Xi in K are
distinct roots of P.
K is unique up to isomorphism and we have Galois
correspondence.
We can prove, for instance, that a P is solvable by radicals if G is
a solvable group.
5 / 25
Let
1
],
i6=j (Xi − Xj )
S = F [X1 , . . . , Xn , Q
maximal in S.
Q
correspondence.
a solvable group.
5 / 25
Let
1
],
i6=j (Xi − Xj )
S = F [X1 , . . . , Xn , Q
maximal in S.
Q
correspondence.
a solvable group.
5 / 25
Let
1
],
i6=j (Xi − Xj )
S = F [X1 , . . . , Xn , Q
maximal in S.
Q
correspondence.
a solvable group.
5 / 25
Let
1
],
i6=j (Xi − Xj )
S = F [X1 , . . . , Xn , Q
maximal in S.
Q
correspondence.
a solvable group.
5 / 25
Linear differential Galois theory (Picard-Vessiot)
Let y 0 = Ay be a linear differential equation of order n (= dim(A)) with
coefficients in F , a differential field with CF algebraically closed.
Easy Fact
The set of solutions of this equation in L > F an extension of F is a
CF -vector space of dimension at most n.
We would like to obtain a corresponding splitting field for this equation
(i.e. with solution space of full (n) dimension).
6 / 25
Easy Fact
6 / 25
Easy Fact
6 / 25
Let
S = F [Y1,1 , . . . , Yn,n ,
1
]
det(Y )
and define a derivation on S setting Y 0 = AY . Let M be a maximal
differential ideal of S.
Define the Picard-Vessiot ring of the given equation with respect
to F as R = S/M. In this case R is not necessarily a field, but it
can be proven that it is an integral domain. Let K = Q(R) and let
G = Aut∂ (K /F ). This will be the Galois group for our equation.
7 / 25
Let
S = F [Y1,1 , . . . , Yn,n ,
1
]
det(Y )
and define a derivation on S setting Y 0 = AY . Let M be a maximal
differential ideal of S.
Define the Picard-Vessiot ring of the given equation with respect
to F as R = S/M. In this case R is not necessarily a field, but it
can be proven that it is an integral domain. Let K = Q(R) and let
G = Aut∂ (K /F ). This will be the Galois group for our equation.
7 / 25
Some properties of P-V extensions
If K /F is a P-V extension for y 0 = Ay , then:
1
K is unique up to isomorphism.
2
K = F (Z ) for some Z ∈ GLn (K ) such that Z 0 = AZ , and
3
CK = CF .
Moreover!
G < GLn (CF ) is a linear algebraic group (i.e. a zero set in
GLn (CF ) of a system of polynomials over CF ).
Properties (2) and (3) above define the P-V extension for the given
differential equation.
8 / 25
1
2
3
CK = CF .
Moreover!
8 / 25
1
2
3
CK = CF .
Moreover!
8 / 25
Galois correspondence
There is a correspondence between Zariski-closed subgroups H ⊂ G
and intermediate differential subfields between F and K .
Solutions of differential equations (One example)
If G is a solvable group then the corresponding equation can be solved
by a finite sequence of field extensions F = F0 < F1 · · · < Fl = K such
that Fi+1 = Fi (ti ) where either
1
ti0 ∈ Fi (integral), or
2
ti0 /ti ∈ Ki (exponential).
9 / 25
Galois correspondence
There is a correspondence between Zariski-closed subgroups H ⊂ G
and intermediate differential subfields between F and K .
Solutions of differential equations (One example)
If G is a solvable group then the corresponding equation can be solved
by a finite sequence of field extensions F = F0 < F1 · · · < Fl = K such
that Fi+1 = Fi (ti ) where either
1
ti0 ∈ Fi (integral), or
2
ti0 /ti ∈ Ki (exponential).
9 / 25
Model Theory, anywhere?
(Robinson, 1950’s!) There is a model companion of the theory of
differential fields (ch. 0): Differentially Closed Fields (a.k.a. DCF0 .)
DCF0 is stable (it is actually totally transcendental (Blum)). It has
QE and elimination of imaginaries. Also, dcl(A) = hAi.
By stability, the field of constants is a pure algebraically closed
field.
10 / 25
field.
10 / 25
field.
10 / 25
Non-linear differential Galois theory (Kolchin)
Let U be a big saturated model of DCF0 .
Strongly normal extensions
An extension K/F of differential fields is strongly normal if:
1
CF = CK algebraically closed.
2
Something like: K = F (Z ) for some Z ∈ GLn (K ) such that
Z 0 = AZ .
Non-linear Galois theory!
Let K /F an strongly normal extension of differential fields, then:
Gal(K /F ) = Aut∂ (K /F ) is isomorphic to the CF rational points of
an algebraic group over CF .
Galois correspondence, &c.
Q: Equations? A: Logarithmic differential equations!.
11 / 25
1
2
K = F (a0 , . . . , am ) and, for any embedding σ : K → U over F , we
have σ(K ) ⊆ K hCU i.
11 / 25
1
2
11 / 25
1
2
11 / 25
1
2
11 / 25
Internality and definable automorphism groups
In the early 80’s, Hrushovski and Zilber isolated conditions to
guarantee that certain automorphism groups were definable (this, in
essence, makes them visible to model theory).
Internality
Consider p and q possibly partial types over A. We say that p is
q-internal (over A) if there is a set B containing A such that, for every
realization a of p, there is a tuple b of realizations of q such that
a ∈ dcl(Bb).
Fundamental system of solutions
Given p q-internal, we say that a tuple a of realizations of p is a
fundamental system of solutions of p relative to q if there exists u(·, ·),
an A-definable function, such that for any b realizing p, we have that
b = u(a, c) for some tuple c of realisations of q. (Conditions?)
12 / 25
Internality
a ∈ dcl(Bb).
12 / 25
Internality
a ∈ dcl(Bb).
12 / 25
Binding group theorem
Let T be a stable theory and U a big saturated model. Suppose that p
and q are over A and p is q-internal. Suppose also that there is a
fundamental system of solutions of p relative to q. Then the group of
automorphisms of U that fixes q(U) and A pointwise induces a
(type-)definable group of automorphisms on p(U)
Actually...
Stability is not really necessary. Something like this can be proven in a
completely general setting. (Hrushovski 2001)
13 / 25
Binding group theorem
Let T be a stable theory and U a big saturated model. Suppose that p
and q are over A and p is q-internal. Suppose also that there is a
fundamental system of solutions of p relative to q. Then the group of
automorphisms of U that fixes q(U) and A pointwise induces a
(type-)definable group of automorphisms on p(U)
Actually...
Stability is not really necessary. Something like this can be proven in a
completely general setting. (Hrushovski 2001)
13 / 25
Poizat’s beautiful observation
Kolchin’s differential Galois Theory could be remade as an application
of internality in DCF0 !
14 / 25
1
2
Corollary (of sorts)
Kolchin’s differential Galois theory
In addition
Motivation for elimination of imaginaries and key question regarding
definable groups in algebraic closed fields!!
15 / 25
1
2
K = F (a) and for any a 0 ≡F a, we have that a 0 ∈ dcl(FaCU ).
(Thus, tp(a/F ) is CU -internal!)
In addition
15 / 25
1
2
In addition
15 / 25
1
2
In addition
15 / 25
Positive characteristic differential Galois theory?
Immediate problem
Direct translation does not work. Ordinary differential constants are too
weak in characteristic p > 0: any extension creates new constants.
16 / 25
Possible solution: Iterative derivations
Differential iterative rings (Hasse, Schmidt (1937!))
An ID-Ring is a ring R equipped with a sequence D = (∂i : R → R)i<ω
of additive endomorphisms such that:
P
(Derivation) ∂0 = IdR and ∂m (ab) = k +l=m ∂k (a)∂l (b);
(Iterativity) ∂m ∂n = m+n
n ∂m+n ;
Given (R, D), define CR , the ring (field) of constants, as the set of
elements that are killed by the ∂i (i ≥ 1).
17 / 25
Possible solution: Iterative derivations
Differential iterative rings (Hasse, Schmidt (1937!))
An ID-Ring is a ring R equipped with a sequence D = (∂i : R → R)i<ω
of additive endomorphisms such that:
P
(Derivation) ∂0 = IdR and ∂m (ab) = k +l=m ∂k (a)∂l (b);
(Iterativity) ∂m ∂n = m+n
n ∂m+n ;
Given (R, D), define CR , the ring (field) of constants, as the set of
elements that are killed by the ∂i (i ≥ 1).
17 / 25
Model theory pays back its debts
Iterative differential fields are well behaved
The theory of iterative differential fields of characteristic p has a stable,
non superstable model companion, SCHp , with elimination of
imaginaries and quantifier elimination.
Actually
SCHp is just another presentation of the theory of separably closed
fields of characteristic p and imperfection degree 1 (SCFp,1 ).
(Remember Soha Sin’s talk on tuesday!)
Precedent
Pillay (2002), following Hrushovski, found a presentation of linear
iterative differential Galois theory using the model theory of SCHp .
18 / 25
Actually
Precedent
18 / 25
Actually
Precedent
18 / 25
Iterative strongly normal extensions
Let U be a large and highly saturated model of SCHp and let C = CU .
Iterative Strongly Normal Extensions
Let (F , D) < (K , D) an extension of definably closed ID-fields. We say
that K /F is a iterative strongly normal extension if:
1
CF = CK and CF is algebraically closed.
2
K = dcl(Fa) (=Closing Fa under derivations and then under pth
roots).
3
For any σ : K ,→ U embedding over F , σ(K ) ⊆ K hCi
4
acl(F ) ∩ K = dcl(Fd) for some d = (d1 , · · · , dm ).
Let Gal(K /F ) be Aut(K /F ).
19 / 25
1
2
roots).
3
4
19 / 25
1
2
roots).
3
4
19 / 25
1
2
roots).
3
4
19 / 25
A good Galois Group and a Galois correspondence
Some results
If K /F is iterative strongly normal with dcl(Fa) = K , then
1
Gal(K /F ) is isomorphic to the CF -rational points of G, an
algebraic group defined over CF , and
2
There is a Galois correspondence between algebraic subgroups
of G(CF ) and definably closed ID-subfields of K which contain F .
3
There are equations associated to these extensions: iterative
logarithmic differential equations.
20 / 25
Some results
1
2
3
20 / 25
Some results
1
2
3
20 / 25
But wait, there is more!
Generalised strongly normal theory: What if we substitute the
constants by an arbitrary formula? (Explored in characteristic zero
by Pillay, Marker, Süer, &c.) The Galois group is finite-dimensional
differential definable. (Ch. p?)
Difference Galois theory: Automorphisms instead of
derivatives? (Kameski (“partial automorphism” (preserving an
almost arbitrary set of formulas), Singer, Chatzidakis, &c.)
21 / 25
But wait, there is more!
Generalised strongly normal theory: What if we substitute the
constants by an arbitrary formula? (Explored in characteristic zero
by Pillay, Marker, Süer, &c.) The Galois group is finite-dimensional
differential definable. (Ch. p?)
Difference Galois theory: Automorphisms instead of
derivatives? (Kameski (“partial automorphism” (preserving an
almost arbitrary set of formulas), Singer, Chatzidakis, &c.)
21 / 25
Arigato Gozaimasu!
22 / 25
Arigato Gozaimasu!
Unless...
22 / 25
One application I dream of:
Galois theory of p-adic differential equations
Let Cp be the completion of the algebraic closure of the field of
p-adic numbers. (|a| = p−vp (a) )
Consider Cp (z) with | · |Gauss , the Gauss norm, defined by
|am z m + · · · + a0 |Gauss = max |ai |.
Denote by F the completion of Cp (z) with respect to the Gauss
df
is continuous
norm. The differentiation on Cp (z) given by f 7→ dz
with respect to the Gauss norm and thus extends uniquely to ∂F a
differentiation of F.
By a p-adic differential equation we mean a differential equation
on the differential field (F, ∂F ).
23 / 25
df
is continuous
23 / 25
df
is continuous
23 / 25
df
is continuous
23 / 25
Fun facts
1
The field of constants of F is Cp .
2
Its residue field is Fp (z) and, more importantly,
3
1 n
The reduction of ( n!
∂F )n≥0 to the residue field is (∂(n) )n≥0 , the
m−n
standard iterative derivative on Fp (z) given by ∂(n) z m = m
.
n z
24 / 25
Fun facts
1
2
3
1 n
m−n
.
n z
24 / 25
Fun facts
1
2
3
1 n
m−n
.
n z
24 / 25
Natural questions
1
What logarithmic differential equations on (F, ∂F ) become iterative
logarithmic differential equations on (Fp (z), ∂(n) ) after reduction?
2
In the cases where this happens, what is the relationship between
the two Galois groups?
Key problem
What is the right model theoretical setting to study structures like
(F, ∂F ) (i.e. differential valued fields of mixed characteristic and the
iterative differential structure in the residue)?
25 / 25
Natural questions
1
2
Key problem
25 / 25
Natural questions
1
2
Key problem
25 / 25

Some applications of model theory to differential Galois theory

Transcription

Similar documents

See attached poster

armedforces

Free Running Differential (FRRD®)

Permazyme Base One

GAL VALLETENZANO (LOCAL ECONOMIC DEVELOPMENT

Quality at the core. canavac.com

brabus 190e 3.6s

3255 AVANTI BAYLINER

FACE-MAKER

special driveline parts for chain drive cars d sport racer c sport racer