b b b b b b b b b b b b b b b b b b b Geometric Singularities of Algebraic Differential Equations Werner M. Seiler ur Mathematik, Universit¨ Institut f¨ at Kassel (joint work with Markus Lange-Hegermann , RWTH Aachen) W.M. Seiler: Geometric Singularities of Algebraic Differential Equations – 1
b b b b b b b b Introduction Introduction Algebraic Differential Equations b b Vessiot Distribution and singularities of differential equations Generalised Solutions Regular Differential Equations � = Geometric Singularities Thomas Decomposition singularities of solutions of differential equations Detection of Singularities related, but different topics � no discussion of shocks, blow-ups, etc. � W.M. Seiler: Geometric Singularities of Algebraic Differential Equations – 2
b b b b b b b b Introduction Introduction “Interpolation” between three domains: Algebraic Differential Equations b b differential algebra � Vessiot Distribution and Generalised Solutions differential topology � Regular Differential differential algebraic equations Equations � Geometric Singularities together with techniques from (differential) algebraic geometry Thomas Decomposition Detection of Singularities Current goal: detect all singularities of given system of (ordinary or partial) differential equations W.M. Seiler: Geometric Singularities of Algebraic Differential Equations – 2
b b b b b b b b Introduction Introduction Differential Algebra Algebraic Differential Equations b b Vessiot Distribution and here mainly differential ideal theory � Generalised Solutions Regular Differential covers automatically systems and all orders � Equations founded by Ritt in early 20th century Geometric Singularities � Thomas Decomposition central goal: understanding singular integrals � Detection of Singularities oldest example: Taylor (1715) � best known example: Clairaut equation (1734) � u = xu ′ + f ( u ′ ) mit f ′′ ( z ) � = 0 ∀ z u ( x ) = cx + f ( c ) general integral: singular integral: x ( τ ) = − f ′ ( τ ) , u ( τ ) = − τf ′ ( τ ) + f ( τ ) W.M. Seiler: Geometric Singularities of Algebraic Differential Equations – 2
b b b b b b b b Introduction Introduction Differential Algebra Algebraic Differential Equations b b Vessiot Distribution and Generalised Solutions Regular Differential Equations Geometric Singularities Thomas Decomposition Detection of Singularities f ( z ) = − 1 4 z 2 W.M. Seiler: Geometric Singularities of Algebraic Differential Equations – 2
b b b b b b b b Introduction Introduction Differential Topology Algebraic Differential Equations b b Vessiot Distribution and singularities of smooth maps between manifolds � Generalised Solutions Regular Differential submanifolds of jet bundles provide geometric model for differential � Equations equations Geometric Singularities Thomas Decomposition � natural projections between jet bundles of different order � Detection of critical points = geometric singularities 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 � W.M. Seiler: Geometric Singularities of Algebraic Differential Equations – 2
b b b b b b b b Introduction Introduction Differential Topology Algebraic Differential Equations b b Vessiot Distribution and Generalised Solutions Regular Differential Equations Geometric Singularities Thomas Decomposition Detection of Singularities ( u ′ ) 2 + u 2 + x 2 = 1 W.M. Seiler: Geometric Singularities of Algebraic Differential Equations – 2
b b b b b b b b Introduction Introduction Differential Algebraic Equations Algebraic Differential Equations b b mainly analytic theory of quasi-linear systems A ( x, u ) u ′ = F ( x, u ) Vessiot Distribution and � Generalised Solutions Regular Differential (including very large systems!) Equations A not necessarily of maximal rank and rank may jump Geometric Singularities � Thomas Decomposition impasse points already discussed in 1960s by electrical engineers � Detection of Singularities � lead to jump phenomena in solutions on one side interpreted as sign of bad model. . . � . . . on the other side experiments often show similar behaviour � W.M. Seiler: Geometric Singularities of Algebraic Differential Equations – 2
b b b b b b b b b b b b b b b b b b b Algebraic Differential Equations C n → C m , u = f ( z ) Introduction consider holomorphic function f : U ⊆ � Algebraic Differential from now on: n , m fixed, U ignored Equations Vessiot Distribution and q -jet [ f ] ( q ) � equivalence class of all holomorphic functions � Generalised Solutions z C n → Regular Differential C m with same Taylor polynomial of degree q around g : Equations z ∈ U as f Geometric Singularities set of all q -jets [ f ] ( q ) Thomas Decomposition jet bundle J q = J q ( C n , C m ) � � z Detection of Singularities � n + q � manifold of dimension d q = n + m � q C d q ) (may be identified with local coordinates ( z , u ( q ) ) corresponding to expansion point z � and derivatives up to order q natural projections for 0 ≤ r < q � � � C n J q − → J r J q − → π q : π q r : [ f ] ( q ) → [ f ] ( r ) [ f ] ( q ) �− �− → z z z z W.M. Seiler: Geometric Singularities of Algebraic Differential Equations – 3
b b b b b b b b b b b b b b b b b b b Algebraic Differential Equations Introduction Definition: Algebraic Differential Equations locally Zariski closed set R q ⊆ J q algebraic jet set of order q � � Vessiot Distribution and Generalised Solutions (i.e.: difference of two varieties) Regular Differential algebraic differential equation of order q � Equations � Geometric Singularities algebraic jet set R q ⊆ J q such that restricted projection π q | R q Thomas Decomposition C n ) dominant (i.e.: image Zariski dense in Detection of Singularities both generalisation and restriction of classical geometric definition: only polynomial non-linearities admitted � R q may have algebraic singularities � equations and inequations admitted � C n dominance replaces surjectivity , permits “special points” in � π q | R q not necessarily submersive � W.M. Seiler: Geometric Singularities of Algebraic Differential Equations – 3
b b b b b b b b b b b b b b b b b b b Algebraic Differential Equations Introduction holomorphic function f defines section Algebraic Differential Equations C n → C n × C m = J 0 , z �→ = [ f ] (0) Vessiot Distribution and � � σ f : z , f ( z ) z Generalised Solutions Regular Differential Equations (graph of f is image of σ f ) Geometric Singularities Thomas Decomposition consider prolonged section Detection of Singularities C n → J q , z �→ [ f ] ( q ) j q σ f : z f (resp. σ f ) (classical) solution of differential equation R q ⊆ J q Def: im ( j q σ f ) ⊆ R q � W.M. Seiler: Geometric Singularities of Algebraic Differential Equations – 3
b b b b b b b b b b b b Vessiot Distribution and Generalised Solutions Introduction C d q ? What distinguishes J q from contact structure on J q � Algebraic Differential Equations Vessiot Distribution and contact distribution C q ⊂ T J q generated by vector fields Def: Generalised Solutions Regular Differential Equations C ( q ) � � u α 1 ≤ i ≤ n = ∂ z i + µ +1 i ∂ u α Geometric Singularities i µ Thomas Decomposition α 0 ≤| µ | <q Detection of Singularities C µ α = ∂ u α 1 ≤ α ≤ m, | µ | = q µ C n → J q of the form γ = j q σ f for function f section γ : Prop: ⇐ ⇒ T im( γ ) ⊂ C q Proof: chain rule! W.M. Seiler: Geometric Singularities of Algebraic Differential Equations – 4
b b b b b b b b b b b b Vessiot Distribution and Generalised Solutions consider prolonged solution j q σ f of equation R q ⊆ J q : Introduction Algebraic Differential Equations � � ur ρ ∈ im( j q σ f ) T ρ im( j q σ f ) � integral elements f¨ � Vessiot Distribution and � � Generalised Solutions solution of R q ⇒ ⊆ T ρ R q = T ρ im( j q σ f ) � Regular Differential � � = ⇒ T ρ im( j q σ f ) ⊆ C q | ρ Equations prolonged section � Geometric Singularities Thomas Decomposition Detection of Singularities W.M. Seiler: Geometric Singularities of Algebraic Differential Equations – 4
b b b b b b b b b b b b Vessiot Distribution and Generalised Solutions consider prolonged solution j q σ f of equation R q ⊆ J q : Introduction Algebraic Differential Equations � � ur ρ ∈ im( j q σ f ) T ρ im( j q σ f ) � integral elements f¨ � Vessiot Distribution and � � Generalised Solutions solution of R q ⇒ ⊆ T ρ R q = T ρ im( j q σ f ) � Regular Differential � � = ⇒ T ρ im( j q σ f ) ⊆ C q | ρ Equations prolonged section � Geometric Singularities Thomas Decomposition Detection of Vessiot space in point ρ on algebraic jet set R q Def: Singularities V ρ [ R q ] = T ρ R q ∩ C q | ρ dim V ρ [ R q ] generally depends on ρ � � regular distribution only on Zariski open subset of R q computing Vessiot distribution V [ R q ] requires only linear algebra � W.M. Seiler: Geometric Singularities of Algebraic Differential Equations – 4
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