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W.M. Seiler: Geometric Singularities of Algebraic Differential Equations – 1

Geometric Singularities of Algebraic Differential Equations

Werner M. Seiler Institut f¨ ur Mathematik, Universit¨ at Kassel (joint work with Markus Lange-Hegermann, RWTH Aachen)

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Introduction

Introduction Algebraic Differential Equations Vessiot Distribution and Generalised Solutions Regular Differential Equations Geometric Singularities Thomas Decomposition Detection of Singularities

W.M. Seiler: Geometric Singularities of Algebraic Differential Equations – 2

singularities of differential equations

=

singularities of solutions of differential equations

related, but different topics
no discussion of shocks, blow-ups, etc.

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Introduction

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W.M. Seiler: Geometric Singularities of Algebraic Differential Equations – 2

“Interpolation” between three domains:

differential algebra
differential topology
differential algebraic equations

together with techniques from (differential) algebraic geometry Current goal: detect all singularities of given system of (ordinary or partial) differential equations

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Introduction

Introduction Algebraic Differential Equations Vessiot Distribution and Generalised Solutions Regular Differential Equations Geometric Singularities Thomas Decomposition Detection of Singularities

W.M. Seiler: Geometric Singularities of Algebraic Differential Equations – 2

Differential Algebra

here mainly differential ideal theory
covers automatically systems and all orders
founded by Ritt in early 20th century
central goal: understanding singular integrals
ldest example:

Taylor (1715)

best known example:

Clairaut equation (1734)

u = xu′ + f(u′)

mit f ′′(z) = 0 ∀z general integral:

u(x) = cx + f(c)

singular integral:

x(τ) = −f ′(τ), u(τ) = −τf ′(τ) + f(τ)

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Introduction

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W.M. Seiler: Geometric Singularities of Algebraic Differential Equations – 2

Differential Algebra

f(z) = −1

4z2

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Introduction

Introduction Algebraic Differential Equations Vessiot Distribution and Generalised Solutions Regular Differential Equations Geometric Singularities Thomas Decomposition Detection of Singularities

W.M. Seiler: Geometric Singularities of Algebraic Differential Equations – 2

Differential Topology

singularities of smooth maps between manifolds
submanifolds of jet bundles provide geometric model for differential

equations

natural projections between jet bundles of different order
critical points = geometric singularities
distinction regular and irregular singularities
complete classifications of singularities of scalar ordinary differential

equations of first or second order

hardly any works on (general) systems

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Introduction

Introduction Algebraic Differential Equations Vessiot Distribution and Generalised Solutions Regular Differential Equations Geometric Singularities Thomas Decomposition Detection of Singularities

W.M. Seiler: Geometric Singularities of Algebraic Differential Equations – 2

Differential Topology

(u′)2 + u2 + x2 = 1

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Introduction

Introduction Algebraic Differential Equations Vessiot Distribution and Generalised Solutions Regular Differential Equations Geometric Singularities Thomas Decomposition Detection of Singularities

W.M. Seiler: Geometric Singularities of Algebraic Differential Equations – 2

Differential Algebraic Equations

mainly analytic theory of quasi-linear systems A(x, u)u′ = F(x, u)

(including very large systems!)

A not necessarily of maximal rank and rank may jump
impasse points already discussed in 1960s by electrical engineers
lead to jump phenomena in solutions
n one side interpreted as sign of bad model. . .
. . . on the other side experiments often show similar behaviour

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Algebraic Differential Equations

Introduction Algebraic Differential Equations Vessiot Distribution and Generalised Solutions Regular Differential Equations Geometric Singularities Thomas Decomposition Detection of Singularities

W.M. Seiler: Geometric Singularities of Algebraic Differential Equations – 3

consider holomorphic function f : U ⊆

Cn → Cm, u = f(z)

from now on: n, m fixed, U ignored

q-jet [f](q)

z

equivalence class of all holomorphic functions

g :

Cn → Cm with same Taylor polynomial of degree q around

z ∈ U as f

jet bundle Jq = Jq( Cn,

Cm)

set of all q-jets [f](q)

z

manifold of dimension dq = n + m

n+q

q

(may be identified with

Cdq)

local coordinates (z, u(q)) corresponding to expansion point z

and derivatives up to order q

natural projections for 0 ≤ r < q

πq

r :

Jq

− → Jr [f](q)

z

− → [f](r)

z

πq :

Jq

− →

Cn

[f](q)

z

− → z

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Algebraic Differential Equations

Introduction Algebraic Differential Equations Vessiot Distribution and Generalised Solutions Regular Differential Equations Geometric Singularities Thomas Decomposition Detection of Singularities

W.M. Seiler: Geometric Singularities of Algebraic Differential Equations – 3

Definition:

algebraic jet set of order q
locally Zariski closed set Rq ⊆ Jq

(i.e.: difference of two varieties)

algebraic differential equation of order q
algebraic jet set Rq ⊆ Jq such that restricted projection πq|Rq

dominant (i.e.: image Zariski dense in

Cn)

both generalisation and restriction of classical geometric definition:

nly polynomial non-linearities admitted
Rq may have algebraic singularities
equations and inequations admitted
dominance replaces surjectivity, permits “special points” in

Cn

πq|Rq not necessarily submersive

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Algebraic Differential Equations

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W.M. Seiler: Geometric Singularities of Algebraic Differential Equations – 3

holomorphic function f defines section

σf :

Cn → Cn × Cm = J0, z →

z, f(z)
= [f](0)

z

(graph of f is image of σf ) consider prolonged section

jqσf :

Cn → Jq, z → [f](q)

z

Def:

f (resp. σf ) (classical) solution of differential equation Rq ⊆ Jq

im (jqσf) ⊆ Rq

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W.M. Seiler: Geometric Singularities of Algebraic Differential Equations – 4

What distinguishes Jq from

Cdq?

contact structure on Jq

Def: contact distribution Cq ⊂ TJq generated by vector fields

C(q)

i

= ∂zi +

α
0≤|µ|<q

uα

µ+1i∂uα

µ

1 ≤ i ≤ n Cµ

α = ∂uα

µ

1 ≤ α ≤ m, |µ| = q

Prop: section γ :

Cn → Jq of the form γ = jqσf for function f

⇐ ⇒ Tim(γ) ⊂ Cq

Proof: chain rule!

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b Vessiot Distribution and Generalised Solutions

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W.M. Seiler: Geometric Singularities of Algebraic Differential Equations – 4

consider prolonged solution jqσf of equation Rq ⊆ Jq:

integral elements
Tρ
im(jqσf)
f¨

ur ρ ∈ im(jqσf)

solution of Rq

= ⇒ Tρ

im(jqσf)
⊆ TρRq
prolonged section

= ⇒ Tρ

im(jqσf)
⊆ Cq|ρ

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W.M. Seiler: Geometric Singularities of Algebraic Differential Equations – 4

consider prolonged solution jqσf of equation Rq ⊆ Jq:

integral elements
Tρ
im(jqσf)
f¨

ur ρ ∈ im(jqσf)

solution of Rq

= ⇒ Tρ

im(jqσf)
⊆ TρRq
prolonged section

= ⇒ Tρ

im(jqσf)
⊆ Cq|ρ

Def: Vessiot space in point ρ on algebraic jet set Rq

Vρ[Rq] = TρRq ∩ Cq|ρ

dim Vρ[Rq] generally depends on ρ
regular distribution only on Zariski open subset of Rq
computing Vessiot distribution V[Rq] requires only linear algebra

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b Vessiot Distribution and Generalised Solutions

Introduction Algebraic Differential Equations Vessiot Distribution and Generalised Solutions Regular Differential Equations Geometric Singularities Thomas Decomposition Detection of Singularities

W.M. Seiler: Geometric Singularities of Algebraic Differential Equations – 4

consider prolonged solution jqσf of equation Rq ⊆ Jq:

integral elements
Tρ
im(jqσf)
f¨

ur ρ ∈ im(jqσf)

solution of Rq

= ⇒ Tρ

im(jqσf)
⊆ TρRq
prolonged section

= ⇒ Tρ

im(jqσf)
⊆ Cq|ρ

Def: Vessiot space in point ρ on algebraic jet set Rq

Vρ[Rq] = TρRq ∩ Cq|ρ

(geometric) symbol:

Nq,ρ = TρRq ∩ Vρπq

q−1 ⊆ V[Rq]

decompose V[Rq] = Nq ⊕ H with complement H (non-unique)
if dim H = n
H horizontal space of Vessiot connection
if Vessiot connection flat (i. e. differential equation integrable)
integral manifolds images of prolonged sections

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b Vessiot Distribution and Generalised Solutions

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W.M. Seiler: Geometric Singularities of Algebraic Differential Equations – 4

Def: differential equation Rq ⊆ Jq

generalised solution
n-dimensional integral manifold N ⊆ Rq
f Vessiot distribution V[Rq]
geometric solution
projection πq

0(N) of generalised solution

N ⊆ Rq

comparison with classical solutions:

geometric solution not necessarily graph of function f
geometric solution πq

0(N) graph of classical solution

⇐ ⇒ N everywhere transversal to πq

geometric solution allow for modelling of multivalued solutions

(“breaking waves”)

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W.M. Seiler: Geometric Singularities of Algebraic Differential Equations – 4

Vessiot distribution and generalised solutions for sphere example

(u′)2 + u2 + x2 = 1

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Regular Differential Equations

Introduction Algebraic Differential Equations Vessiot Distribution and Generalised Solutions Regular Differential Equations Geometric Singularities Thomas Decomposition Detection of Singularities

W.M. Seiler: Geometric Singularities of Algebraic Differential Equations – 5

P =

C[x1, . . . , xn] polynomial ring in n variables, point ρ ∈ Jq

Φ : Jq →

C holomorphic function; ¯

q maximal order of jet variable uα

µ

actually appearing in Φ

principal part of Φ in ρ

ppρ Φ =

m

α=1
|µ|=¯

q

∂Φ ∂uα

µ

(ρ)xµeα ∈ Pm Rq described by equations Φ1 = 0, . . . , Φr = 0 (inequations irrelevant)

in every point ρ ∈ Rq polynomial module

M[ρ] = ppρ Φ1, . . . , ppρ Φr

Def: Hilbert function of Rq in ρ

Hilbert function of factor module Pm/M[ρ]

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Regular Differential Equations

Introduction Algebraic Differential Equations Vessiot Distribution and Generalised Solutions Regular Differential Equations Geometric Singularities Thomas Decomposition Detection of Singularities

W.M. Seiler: Geometric Singularities of Algebraic Differential Equations – 5

Def: algebraic differential equation Rq ⊆ Jq regular

Rq smooth (i.e. manifold)
Hilbert functions independent of point ρ ∈ Rq

idea: uniform behaviour of all “characteristic values” over Rq, in particular

dim Vρ[Rq]
dim Nq,ρ
size of formal solution space in ρ

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W.M. Seiler: Geometric Singularities of Algebraic Differential Equations – 6

consider algebraic differential equation Rq ⊆ Jq

algebraic singularities
singularities in the sense of algebraic

geometry

ignored in sequel (not much known)
determination classical problem in algebraic geometry

(Jacobi criterion)

geometric singularities
critical points of restricted projection

ˆ πq

q−1 : Rq −

→ πq

q−1(Rq)

(i.e. Tρˆ

πq

q−1 not surjective)

points where dimension of symbol jumps

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let Rq ⊆ Jq be union of algebraic jet sets; smooth point ρ ∈ Rq is

regular
ρ has open neighbourhood where V[Rq] regular and

V[Rq] = Nq ⊕ H with dim H = n

regular singular
ρ has open neighbourhood where V[Rq]

regular, but dim Hρ < n

irregular singular
V[Rq] not regular on any neighbourhood of ρ
purely irregular singular
irregular singularity with dim Hρ = n

difference to classical definitions:

partial differential equations require consideration of neighbourhood of

point

“right” dimension of V[Rq] a priori not known
for ordinary differential equations pointwise analysis sufficient

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W.M. Seiler: Geometric Singularities of Algebraic Differential Equations – 6

formally integrable ODE:

Rq ⊆ Jq

local description:

Φ(z, u(q)) = 0 Rq of finite type

almost everywhere dim Vρ[Rq] = 1

Prop: point ρ ∈ Rq

regular

⇐ ⇒ rank

C(q)Φ
ρ = m
regular singular

⇐ ⇒ ρ not regular and rank

C(q)Φ | C(q)

transΦ

ρ = m

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W.M. Seiler: Geometric Singularities of Algebraic Differential Equations – 6

formally integrable ODE:

Rq ⊆ Jq

local description:

Φ(z, u(q)) = 0 Rq of finite type

almost everywhere dim Vρ[Rq] = 1

Thm: assume Rq without irregular singularities

ρ ∈ Rq regular point

= ⇒

(i) unique classical solution f exists with ρ ∈ im jqσf (ii) solution f may be continued in any direction until jqσf reaches either boundary of Rq or a regular singularity

ρ ∈ Rq regular singularity

= ⇒

dichotomy (i) either two classical solutions f1, f2 exist with ρ ∈ im jqσfi (either both start or both end in ρ) (ii)

r one classical solution f exists with ρ ∈ im jqσf whose

derivative of order q + 1 in z = πq(ρ) is not defined

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W.M. Seiler: Geometric Singularities of Algebraic Differential Equations – 6

formally integrable ODE:

Rq ⊆ Jq

local description:

Φ(z, u(q)) = 0 Rq of finite type

almost everywhere dim Vρ[Rq] = 1

Proof:

V[Rq] locally generated by vector field X ρ regular singularity = ⇒ X vertical wrt πq

dichotomy

does ∂z-component of X change sign in ρ?

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W.M. Seiler: Geometric Singularities of Algebraic Differential Equations – 6

formally integrable ODE:

Rq ⊆ Jq

local description:

Φ(z, u(q)) = 0 Rq of finite type

almost everywhere dim Vρ[Rq] = 1

let ρ ∈ Rq be an irregular singularity

consider simply connected domain U ⊂ Rq without irregular

singularities such that ρ ∈ U

in U Vessiot distribution V[Rq] generated by vector field X

Thm: generically every continuation of X to ρ vanishes Consequence: solution behaviour in neighbourhood of isolated irregular singularity ρ analysable with dynamical systems theory (mainly determined by eigenstructure of Jacρ X)

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W.M. Seiler: Geometric Singularities of Algebraic Differential Equations – 6

Example:

(u′)3 + uu′ − x = 0

(hyperbolic gather) singularity curve (criminant):

3(u′)2 + u = 0

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W.M. Seiler: Geometric Singularities of Algebraic Differential Equations – 6

Example:

(u′)3 + uu′ − x = 0

(hyperbolic gather) second derivative of solution touching “tip” of discriminant does not exist solutions “change direction” when crossing discriminant

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W.M. Seiler: Geometric Singularities of Algebraic Differential Equations – 6

Example:

(u′)3 + uu′ − x = 0

(hyperbolic gather) neighbourhood of irregular singularity the two solutions tangential to eigenvectors of Jac X intersect

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Thomas Decomposition

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W.M. Seiler: Geometric Singularities of Algebraic Differential Equations – 7

Algebraic Case polynomial ring

C[z1, . . . , zn] with total ordering on variables

leader ld p
largest variable in polynomial p
consider p as univariate polynomial in ld p
initial init p
leading coefficient of p
separant sep p
∂p/∂(ld p)

algebraic system

finite set of polynomial equations and inequations

S =

p1 = 0, . . . , ps = 0, q1 = 0, . . . , qt = 0
solution set

(locally closed wrt Zariski topology)

Sol S =

z ∈

Cn | pi(z) = 0, qj(z) = 0

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Thomas Decomposition

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W.M. Seiler: Geometric Singularities of Algebraic Differential Equations – 7

Algebraic Case Def: simple algebraic system

triangular:
{ld pi, ld qj} \ {1}
= s + t
non-vanishing initials:

no equation init pi = 0 or init qj = 0 has solution in Sol S

square-free:

dito for separants Def: Thomas decomposition of algebraic system S

finitely many

simple systems S1, . . . , Sk such that Sol S disjoint union of all Sol Si

exists always
depends on ordering of variables
decomposes according to fibre cardinality for coordinate projections
can be determined algorithmically

(subresultants, case distinctions

expensive!)

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Algebraic Case consider V

y3 + (3x + 1)y2 + (3x2 + 2x)y + x3

x y

Thomas decomposition

S1 =
y3 +(3x+1)y2 +(3x2 +2x)y +x3 = 0, 27x3 −4x = 0
S2 =
6y2 −(27x2 −12x−6)y −3x2 +2x = 0, 27x3 −4x = 0

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Thomas Decomposition

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W.M. Seiler: Geometric Singularities of Algebraic Differential Equations – 7

differential case ring of differential polynomials

K =

C(z1, . . . , zn)

rational functions

derivations

δi = ∂/∂zi

differential unknowns:

U = {u1, . . . , um}

jet variables uα

µ = δµuα

K{U} =

K

uα

µ | 1 ≤ α ≤ m, µ ∈

Nn

(polynomial ring in infinitely many variables)

derivations can be extended:

δiuα

µ = uα µ+1i

distinguish:
algebraic ideal:

p1, . . . , ps

differential ideal:

p1, . . . , ps∆

set D =

C

zi, uα

µ

⊂

K{U}

Dq =

C

zi, uα

µ | |µ| ≤ q

coordinate ring of Jq

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differential case ranking on

K{U}

total ordering ≺ of jet variables
uα ≺ δiuα
uα

µ ≺ uβ ν

= ⇒ δiuα

µ ≺ δiuβ ν

extend concepts like leader, initial or separant differential system

finite set of differential polynomial equations and

inequations

S =

p1 = 0, . . . , ps = 0, q1 = 0, . . . , qt = 0
solution set
consider formal solutions

(different function spaces possible)

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differential case Def: simple differential system

simple as algebraic system in the finitely many occuring jet variables
involutive for Janet division
no leader of inequation derivative of leader of equation

Def: Thomas decomposition of differential system S

finitely many

simple systems S1, . . . , Sk such that Sol S disjoint union of all Sol Si

exists always
algorithmically computable via combination of algebraic Thomas

decomposition and Janet-Riquier theory

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W.M. Seiler: Geometric Singularities of Algebraic Differential Equations – 8

starting point: differential system S goal: all geometric singularities in given order q differential computation:

differential Thomas decomposition (other methods also possible)
simple differential systems Si
ne Si corresponds to general integral

all others yield singular integrals

all other kind of singularities eliminated

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W.M. Seiler: Geometric Singularities of Algebraic Differential Equations – 8

starting point: differential system S goal: all geometric singularities in given order q algebraic analysis of simple differential system S

introduce suitable ideals
I(S) = p1, . . . , ps∆ : h∞ ⊆ D with

h =

i sep (pi) init (pi)

Iq(S) = I(S) ∩ Dq,

Kq = qj | ord(qj) ≤ qDq

algebraic jet set

Rq = Sol

Iq(S)
\ Sol
Kq(S)
⊆ Jq

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W.M. Seiler: Geometric Singularities of Algebraic Differential Equations – 8

starting point: differential system S goal: all geometric singularities in given order q algebraic analysis of simple differential system S

ansatz for Vessiot distribution of Rq

X =

i

aiC(q)

i

+

α,µ

bα

µCµ α

extended polynomial ring DV

q = Dq[a, b] with b ≻ a ≻ u ≻ z

compute algebraic Thomas decomposition of system over DV

q

consisting of generators of Iq(S) plus equations for Vessiot distribution (linear in a, b)

solve “parametric linear system”
simple systems SV

i

and Si = SV

i ∩ Dq

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W.M. Seiler: Geometric Singularities of Algebraic Differential Equations – 8

starting point: differential system S goal: all geometric singularities in given order q Def: regularity decomposition in order q of simple differential system

write Sol Iq(S) ⊆ Jq as disjoint union of finitely many regular

algebraic jet sets R(i)

q

⊂ Jq (components in order q)

singular closure R(i)

q

f component R(i)

q

union with all

components R(j)

q

lying in Zariski closure of R(i)

q

constituent in order q
algebraically simple system S′ such that

Sol (S′) ⊆ Sol

Iq(S)
and set of leaders of equations in S′ equal to

set of jet variables in ld p1, . . . , ld ps∆ ∩ Dq

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Introduction Algebraic Differential Equations Vessiot Distribution and Generalised Solutions Regular Differential Equations Geometric Singularities Thomas Decomposition Detection of Singularities

W.M. Seiler: Geometric Singularities of Algebraic Differential Equations – 8

starting point: differential system S goal: all geometric singularities in given order q final analysis:

Thm:

algorithm yields regularity decomposition with all constituents regular algebraic differential equations

Prop:

union of solution sets of constituents Zariski dense in

Sol

Iq(S)
consider for each constituent singular closure

(component may lie in closure of several constituents!)

taxonomy of singularities via leaders of systems SV

i

(do variables a appear as leader?) and comparison with constituent