b Geometric Singularities of b Algebraic Differential ... · Introduction Algebraic Differential Equations Vessiot Distribution and Generalised Solutions Regular Differential Equations

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

Geometric Singularities of

Algebraic Differential Equations

Werner M. Seiler

Institut fur Mathematik, Universitat 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

6=

singularities of solutions of differential equations

� related, but different topics

� no discussion of shocks, blow-ups, etc.

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“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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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

� oldest example: Taylor (1715)

� best known example: Clairaut equation (1734)

u = xu′ + f(u′) mit f ′′(z) 6= 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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Differential Algebra

f(z) = −14z

2

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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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Differential Topology

(u′)2 + u2 + x2 = 1

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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

� on one side interpreted as sign of bad model. . .

� . . . on the other side experiments often show similar behaviour

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� consider holomorphic function f : U ⊆ Cn → C

m, u = f(z)from now on: n, m fixed, U ignored

� q-jet [f ](q)z equivalence class of all holomorphic functions

g : Cn → C

m with same Taylor polynomial of degree q around

z ∈ U as f

� jet bundle Jq = Jq(C

n,Cm) set of all q-jets [f ](q)z

� manifold of dimension dq = n+m(

n+qq

)

(may be identified withCdq )

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

and derivatives up to order q� natural projections for 0 ≤ r < q

πqr :

{

Jq −→ Jr

[f ](q)z 7−→ [f ]

(r)z

πq :

{

Jq −→ C

n

[f ](q)z 7−→ z

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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 inCn)

both generalisation and restriction of classical geometric definition:

� only polynomial non-linearities admitted

� Rq may have algebraic singularities

� equations and inequations admitted

� dominance replaces surjectivity, permits “special points” inCn

� πq|Rqnot necessarily submersive

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holomorphic function f defines section

σf : C

n → C

n ×Cm = J0, z 7→(

z, f(z))

= [f ](0)z

(graph of f is image of σf )

consider prolonged section

jqσf : C

n → Jq, z 7→ [f ](q)z

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

im (jqσf ) ⊆ Rq

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What distinguishes Jq fromCdq? contact structure on Jq

Def: contact distribution Cq ⊂ TJq generated by vector fields

C(q)i = ∂zi +

∑

α

∑

0≤|µ|<q

uαµ+1i∂uαµ

1 ≤ i ≤ n

Cµα = ∂uα

µ1 ≤ α ≤ m, |µ| = q

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

⇐⇒ T im(γ) ⊂ Cq

Proof: chain rule!

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consider prolonged solution jqσf of equation Rq ⊆ Jq :

� integral elements Tρ

(

im(jqσf ))

fur ρ ∈ im(jqσf )� solution of Rq =⇒ Tρ

(

im(jqσf ))

⊆ TρRq

� prolonged section =⇒ Tρ

(

im(jqσf ))

⊆ Cq|ρ

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(

im(jqσf ))


(

im(jqσf ))

⊆ TρRq


(

im(jqσf ))

⊆ Cq|ρ

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

Vρ[Rq] = TρRq ∩ Cq|ρ

� dimVρ[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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(

im(jqσf ))


(

im(jqσf ))

⊆ TρRq


(

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 ∩ Vρπqq−1 ⊆ V[Rq]

� decompose V[Rq] = Nq ⊕H with complement H (non-unique)

� if dimH = 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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Def: differential equation Rq ⊆ Jq

� generalised solution n-dimensional integral manifold N ⊆ Rq

of 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 πq0(N ) graph of classical solution ⇐⇒

N everywhere transversal to πq

� geometric solution allow for modelling of multivalued solutions

(“breaking waves”)

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Vessiot distribution and generalised solutions for sphere example

(u′)2 + u2 + x2 = 1

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Regular Differential Equations

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

Φ : Jq → C holomorphic function; q maximal order of jet variable uαµactually appearing in Φ principal part of Φ in ρ

ppρΦ =m∑

α=1

∑

|µ|=q

∂Φ

∂uαµ(ρ)xµeα ∈ Pm

Rq described by equations Φ1 = 0, . . . ,Φr = 0 (inequations irrelevant)

in every point ρ ∈ Rq polynomial module

M[ρ] = 〈ppρΦ1, . . . , ppρΦr〉

Def: Hilbert function of Rq in ρ Hilbert function of factor module Pm/M[ρ]

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Regular Differential Equations

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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

� dimVρ[Rq]� dimNq,ρ

� size of formal solution space in ρ

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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

πqq−1 : Rq −→ πq

q−1(Rq) (i.e. Tρπqq−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 dimH = n� regular singular ρ has open neighbourhood where V[Rq]

regular, but dimHρ < n� irregular singular V[Rq] not regular on any neighbourhood of ρ� purely irregular singular irregular singularity with dimHρ = 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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formally integrable ODE: Rq ⊆ Jq

local description: Φ(z,u(q)) = 0Rq of finite type almost everywhere dimVρ[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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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) or 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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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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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 JacρX)

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Example: (u′)3 + uu′ − x = 0 (hyperbolic gather)

singularity curve (criminant):

3(u′)2 + u = 0

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second derivative of solution

touching “tip” of discriminant

does not exist

solutions “change direction”

when crossing discriminant

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neighbourhood of irregular singularity

the two solutions tangential to eigenvectors of JacX intersect

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Algebraic Case

polynomial ringC[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 6= 0, . . . , qt 6= 0}

solution set (locally closed wrt Zariski topology)

SolS ={

z ∈ Cn | pi(z) = 0, qj(z) 6= 0}

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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 SolS� square-free: dito for separants

Def: Thomas decomposition of algebraic system S finitely many

simple systems S1, . . . ,Sk such that SolS disjoint union of all SolSi

� 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 6= 0}

� S2 ={

6y2− (27x2−12x−6)y−3x2+2x = 0, 27x3−4x = 0}

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differential case

ring of differential polynomials

� K = C(z1, . . . , zn) rational functions

� derivations δi = ∂/∂zi� differential unknowns: U = {u1, . . . , um} jet variables uαµ = δµuα

� K{U} = K

[

uαµ | 1 ≤ α ≤ m,µ ∈ Nn0

]

(polynomial ring in infinitely many variables)

derivations can be extended: δiuαµ = uαµ+1i

� distinguish:

� algebraic ideal: 〈p1, . . . , ps〉� differential ideal: 〈p1, . . . , ps〉∆

� set D = C

[

zi, uαµ

]

⊂ K{U} Dq = C

[

zi, uαµ | |µ| ≤ q

]

coordinate ring of Jq

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differential case

ranking onK{U}

� total ordering ≺ of jet variables

� uα ≺ δiuα

� uαµ ≺ uβν =⇒ δiuαµ ≺ δiu

βν

extend concepts like leader, initial or separant

differential system finite set of differential polynomial equations and

inequations

S ={

p1 = 0, . . . , ps = 0, q1 6= 0, . . . , qt 6= 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 SolS disjoint union of all SolSi

� exists always

� algorithmically computable via combination of algebraic Thomas

decomposition and Janet-Riquier theory

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starting point: differential system Sgoal: all geometric singularities in given order q

differential computation:

� differential Thomas decomposition (other methods also possible)

simple differential systems Si

� one Si corresponds to general integral

all others yield singular integrals

� all other kind of singularities eliminated

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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) ≤ q〉Dq

algebraic jet set

Rq = Sol(

Iq(S))

\ Sol(

Kq(S))

⊆ Jq

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algebraic analysis of simple differential system S

� ansatz for Vessiot distribution of Rq

X =∑

i

aiC(q)i +

∑

α,µ

bαµCµα

extended polynomial ring DVq = Dq[a,b] with b ≻ a ≻ u ≻ z

� compute algebraic Thomas decomposition of system over DVq

consisting of generators of Iq(S) plus equations for Vessiot

distribution (linear in a,b) solve “parametric linear system”

simple systems SVi and Si = SV

i ∩ Dq

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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 of 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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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 SVi (do variables a

appear as leader?) and comparison with constituent

b Geometric Singularities of b Algebraic Differential ... · Introduction Algebraic Differential Equations Vessiot Distribution and Generalised Solutions Regular Differential Equations

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