Constructive Algebra for Differential Invariants

Transcription

1 Constructive Algebra for Differential Invariants Evelyne Hubert Rutgers, November 2008 Constructive Algebra for Differential Invariants Differential invariants arise in equivalence problems and are used in symmetry reduction techniques We introduce a computationally relevant differential algebraic structure for those. Constructive Algebra for Differential Invariants Contents 1 Differential Algebra and Related Topics 1 2 Lie Group Actions and their Invariants 2 3 Normalized Invariants: Geometric and Algebraic Construction 5 4 Differential Invariant Derivation, Syzygies 6 5 Bibliography 8 1 Differential Algebra and Related Topics Differential Polynomial Rings δ 1 = x, δ2 = y Y = {φ, ψ} F = Q(x, y) F φ, ψ = F[φ, φ x, φ y,..., ψ...] F a field = {δ 1,..., δ m } derivations on F Y = {y 1,..., y n } F[ y α α N m, y Y ] = F Y φ xxy φ x 2 y φ (2,1) x (φxxy) = φxxxy δ1 `φ(2,1) = φ(3,1) x y = y x δ i (y α ) = y α+ɛi ɛ i = (0,..., 1 i th,..., 0) δ i δ j = δ j δ i Link with differential geometry Independent variables x 1,..., x n F = Q(x 1,..., x m ) Dependent variables y 1,..., y n Y = {y 1,..., y n } the differential indeterminates 1

2 F Y is the coordinate ring for the infinite jet space Total derivatives: δ i = x i + y Y, α N m y α+ɛi y α Derivations with nontrivial commutations Y = {y 1,..., y n } = {δ 1,..., δ m } m δ i δ j δ j δ i = c ijl δ l l=1 c ijl K Y K Y? [H05] Differential polynomial ring K Y with non commuting derivations Y = {y 1,..., y n } D = {δ 1,..., δ m } K[y α α N m, y Y] 8 >< δ i (yα) = >: y α+ɛi ifα 1 =... = α i 1 = 0 «mx δ j δ i y α ɛj + c ijl δ l y α ɛj «l=1 where j < i is s.t. α j > 0 while α 1 =... = α j 1 = 0 If the c ijl satisfy - c ijl = c jil mx - δ k (c ijl ) + δ i (c jkl ) + δ j (c kil ) = c ijµ c µkl + c jkµ c µil + µ=1 c kiµ c µjl m then δ i δ j (p) δ j δ i (p) = c ijl δ l (p) p K[ y α α N m ] = K Y l=1 & there exists an admissible ranking - α < β y α y β, - y α z β y α+γ z β+γ, - c ijl δ l (y α ) y α+ɛi+ɛ j l N m [H05] 2 Lie Group Actions and their Invariants Lie (Algebraic) Group G G a r-dimensional smooth manifold, locally parameterized by R r m : G G G (λ, µ) λ µ e G and i : G G λ λ 1 smooth e λ = λ e = λ 2

3 Group action G a Lie group M an open subset of R n Action g : G M M (λ, z) λ z smooth e z = z (λ µ) z = λ (µ z) Orbit of z: O z = {λ z λ G} M Semi-regular Lie group actions scaling translation+reflection rotation G R R { 1, 1} SO(2) Orbits: M R 2 \ O R 2 R 2 \ O λ z ( λ z1 λ z 2 ) ( z1 + λ 1 λ 2 z 2 ) ( cos λ sin λ ) ( z1 sin λ cos λ z 2 ) Infinitesimal generator x x + y y Infinitesimal generators x ξ 1 y x x y ξ d z 1 z d a vector field the flow of which is the action of a one-dimensional (connected) subgroup of G. V 1,..., V r a basis of infinitesimal generators for the action on M of the r-dimensional group G. Local Invariants f : U M RK smooth f(λ z) = f(z) for λ G close to e f is constant on orbits within U V 1 (f) = 0,..., V r (f) = 0 3

4 Examples G K K { 1, 1} SO(2) rational x y y 2 x 2 + y 2 local x y y x2 + y 2 Classical differential invariants E(2) α 2 + β 2 = 1 ( ) ( ) ( ) ( ) X α β x a = + Y β α y b Y X = β + αy x α β y Y XX = y xx (α β y) 3 Curvature: σ = y 2 xx (1+y 2 x )3 a differential invariant Arc length: ds = 1 + y 2 x dx Invariant derivation: d ds = y 2 x d dx Jets / Differential algebraprolongation J 0 = X U g (0) : G J 0 J 0 V 0 1,..., V 0 r (x 1,..., x m ) coordinates on X independent variables (u 1,..., u n ) coordinates on U dependent variables J k = X U (k) g (k) : G J k J k V k 1,..., V k r [DifferentialGeometry] 4

5 additional coordinates u α = α u x α, α k the derivatives of u w.r.t x up to order k Differential polynomial ring: K(x) u = K(x) [u α α N m ] D i u α = u α+ɛi. [diffalg] f : J k R differential invariant of order k if V k (f) = 0. Invariant derivation D : F(J k ) F(J k+1 ) s.t D V = V D f : J k R a differential invariant D(f) a differential invariant of order k + 1. What is a computationnally relevant algebraic structure for differential invariants? K y 1,..., y n / S 3 Normalized Invariants: Geometric and Algebraic Construction Local cross-section P P P an embedded manifold of dimension n d P = {z U p 1 (z) =... = p d (z) = 0} z z O z P is transverse to O z at z P. P intersect O 0 z at a unique point, z U. the matrix (V i (p j )) 1 i r,1 j d has rank d on P. A local invariant is uniquely determined by a function on P. [Fels Olver 99, H. Kogan 07b] Invariantization ῑf of a function f P f : U R smooth z z O z ῑf is the unique local invariant with ῑf P = f P ῑf(z) = f ( z) Normalized invariants: ῑz 1,..., ῑz n. ῑf(z) = f(ῑz) Generation and rewriting: f local invariant f(z 1,..., z n ) = f(ῑz 1,..., ῑz n ) Relations: p 1 (ῑz 1,..., ῑz n ) = 0,..., p d (ῑz 1,..., ῑz n ) = 0 [Fels Olver 99, H. Kogan 07b] 5

6 Normalized invariants. Example. G = SO(2), M = R 2 \ O z P : z 2 = 0, z 1 > 0 U = M P ) (ῑz 1, ῑz 2 ) = ( z 21 + z22, 0 Replacement property: f(z 1, z 2 ) invariant f(z 1, z 2 ) = f (ῑz 1, 0). Normalized invariants in practice We mostly do not need (ῑz 1,..., ῑz n ) explicitly. We can work formally with (ῑz 1,..., ῑz n ), subject to the relationships p 1 (ῑz) = 0,..., p d (ῑz) = 0. Computing normalized invariants In the algebraic case, the normalized invariants (ῑz 1,..., ῑz n ) for a K(z) G -zero of the graph-section ideal (G + (Z λ z) + P ) K(z)[Z] The coefficients of the reduced Gröbner basis of the graph-section ideal form a generating set for K(z) G endowed with a simple rewriting algorithm. [H. Kogan 07a 07b] 4 Differential Invariant Derivation, Syzygies Differential invariants J 0 = X U g (0) : G J 0 J 0 V 0 1,..., V 0 r (x 1,..., x m ) coordinates on X (u 1,..., u n ) coordinates on U J k = X U (k) g (k) : G J k J k V k 1,..., V k r additional coordinates u α = α u x α, α k Normalized invariants of order k I k = {ῑx 1,..., ῑx m } {ῑu α α k} 6

7 Generation in finite terms r k, the dimension of orbits on J k, stabilizes at order s r 0 r 1... r s = r s+1 =... = r. P s : p 1 = 0,..., p r = 0 defines a cross-section on J s+k I s+k = {ῑx 1,..., ῑx m } {ῑu α α s + k} Construct: D 1,..., D m : F(J s+k ) F(J s+k+1 ) s.t. D i V a = V a D i Key Prop: ῑu α+ɛi = D i (ῑu α ) + K ia ῑ (V a (u α )) K = ῑ ( D(P ) V(P ) 1) Col: Any differential invariants can be contructively written in terms of I s+1 and their derivatives. Algebra of Differential Invariants [Fels Olver 99] K x i, u α α s + 1 / S [ ] m Di, D j = Λ ijk D k k=1 where Λ ijk = P r c=1 Kic ῑ(d j(v c (x k ))) K jc ῑ(d i (V c (x k ))). { The monotone derivatives } { of I s+1, } D β1 1...Dβm m (ῑx i ) D β1 1...Dβm m (ῑu α ) α s + 1, generate all differential invariants. Syzygies=Differential relationships A subset S of the following relationships [H 05, 08] p 1 (ῑx, ῑu α ) = 0,..., p r (ῑx, ῑu α ) = 0 D i (ῑx j ) = δ ij K ia ῑ (V(x j )), D i (ῑu α ) = ῑu α+ɛi K ia ῑ (V(u α )), α s D i (ῑu α ) D j (ῑu β ) = K ja ῑ (V(u β )) K ia ῑ (V(u α )), α + ɛ i = β + ɛ j, α = β = s + 1. form a complete set of differential syzygies. ιu yy ῑu α+ɛi = D i (ῑu α ) + K ia ῑ (V a (u α )) ιu y ιu [H08] ιu x ιu xx s + 1 K = ῑ ( D(P ) V(P ) 1) 7

8 Smaller set of generators Differential elimination on a complete set of syzygies allows to reduce the number of differential invariants. We can reduce to two or even 1 the number of generating invariants by differential elimination on the syzygies. Euclidean and affine surfaces [Olver 07] Conformal and projective surfaces [H. Olver 07] We can always restrict to m r + d 0 generating invariants ῑu α+ɛi = D i (ῑu α ) + K ia ῑ (V a (u α )) K = ῑ ( D(P ) V(P ) 1) The edge invariants E = {ῑ(d i (p a ))} I 0 form a generating set when the the cross-section is of minimal order. We can obtain their syzygies by elimination. The Maurer-Cartan invariants {K ia } I 0 form a generating set of differential invariants. We can obtain their syzygies from the structure equations. Merci. Thanks. 5 Bibliography Origins of the project: Symmetry reduction [H07, H08] [1] E. Mansfield. Algorithms for symmetric differential systems. Foundations of Computational Mathematics, 1(4) (20), [2] M. Fels and P. J. Olver. Moving coframes: II. Regularization and theoretical foundations. Acta Applicandae Mathematicae, 55(2) (1999), Computing (algebraic) invariants [1] E. Hubert and I. A. Kogan. Rational invariants of a group action. Construction and rewriting. J. of Symbolic Computation, 42(1-2): , [2] E. Hubert and I. A. Kogan. Smooth and algebraic invariants of a group action. Local and global constructions. Foundations of Computational Mathematics, 7(4) (2007), Differential Algebraic Structure of Invariants [1] E. Hubert. Differential invariants of a Lie group action: syzygies on a generating set. Journal of Symbolic Computation, [2] E. Hubert. Generation properties of Maurer-Cartan invariants Computing in the Algebra of Differential Invariants 8

9 [1] E. Hubert. Differential algebra for derivations with nontrivial commutation rules. Journal of Pure and Applied Algebra, 200(1-2) (2005), [2] E. Hubert and P. J. Olver. Differential invariants of conformal and projective surfaces. Symmetry Integrability and Geometry: Methods and Applications, 3 (2007), Art Software [1] E. Hubert. diffalg: extension to non commuting derivations. INRIA, Hubert/diffalg. [2] E. Hubert. aida - Algebraic Invariants and their Differential Algebra. INRIA, Hubert/aida. [3] I. Anderson DifferentialGeometry - previously Vessiot. Maple 11. 9