Post on 19-Jun-2020
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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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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
“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
� 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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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
f(z) = −14z
2
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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
� on 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 → 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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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 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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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
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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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
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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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 ))
fur ρ ∈ im(jqσf )� solution of Rq =⇒ Tρ
(
im(jqσf ))
⊆ TρRq
� prolonged section =⇒ Tρ
(
im(jqσf ))
⊆ Cq|ρ
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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 ))
fur 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
� 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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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 ))
fur 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 ∩ 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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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
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
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
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 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
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
� dimVρ[Rq]� dimNq,ρ
� size of formal solution space in ρ
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Geometric Singularities
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 – 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
πqq−1 : Rq −→ πq
q−1(Rq) (i.e. Tρπqq−1 not surjective)
points where dimension of symbol jumps
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Geometric Singularities
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 – 6
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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Geometric Singularities
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 – 6
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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Geometric Singularities
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 – 6
formally integrable ODE: Rq ⊆ Jq
local description: Φ(z,u(q)) = 0Rq of finite type almost everywhere dimVρ[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) 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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Geometric Singularities
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 – 6
formally integrable ODE: Rq ⊆ Jq
local description: Φ(z,u(q)) = 0Rq of finite type almost everywhere dimVρ[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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Geometric Singularities
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 – 6
formally integrable ODE: Rq ⊆ Jq
local description: Φ(z,u(q)) = 0Rq of finite type almost everywhere dimVρ[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 JacρX)
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Geometric Singularities
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 – 6
Example: (u′)3 + uu′ − x = 0 (hyperbolic gather)
singularity curve (criminant):
3(u′)2 + u = 0
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Geometric Singularities
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 – 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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Geometric Singularities
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 – 6
Example: (u′)3 + uu′ − x = 0 (hyperbolic gather)
neighbourhood of irregular singularity
the two solutions tangential to eigenvectors of JacX intersect
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Thomas Decomposition
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 – 7
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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Thomas Decomposition
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 – 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 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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Thomas Decomposition
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 – 7
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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Thomas Decomposition
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 – 7
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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Thomas Decomposition
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 – 7
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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Thomas Decomposition
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 – 7
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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Detection of Singularities
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 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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Detection of Singularities
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 Sgoal: 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) ≤ q〉Dq
algebraic jet set
Rq = Sol(
Iq(S))
\ Sol(
Kq(S))
⊆ Jq
b
bb
bb
b
b
b
b
Detection of Singularities
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 Sgoal: 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
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
b
bb
bb
b
b
b
b
Detection of Singularities
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 Sgoal: 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 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
b
bb
bb
b
b
b
b
Detection of Singularities
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 Sgoal: 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 SVi (do variables a
appear as leader?) and comparison with constituent