Algebraic and Differential Invariants

Evelyne HubertINRIA, Mediterranee

www.inria.fr/members/Evelyne.Hubert

Budapest - FoCM 2011

Joint work and shared thoughts with:Elizabeth Mansfield, Irina Kogan, Gloria Mari-BeffaPeter Olver, Michael Singer,George Labahn,Peter van der Kamp, Mark Hickman.

Contents

1 Scaling symmetries 2

2 Rational and algebraic invariants 4

3 Differential Invariants 8

4 Generalized Differential Algebra 12

Referencesa

M. Fels and P. J. Olver. Moving coframes: II. Regularization and theoretical foundations. Acta Applicandae Mathematicae,55(2) (1999).

aP. J. Olver. Generating differential invariants. Journal of Mathematical Analysis and Applications 333 (2007).

aE. Mansfield,P. H. van der Kamp. Evolution of curvature invariants and lifting integrability J. Geom. Phys. 56:8 (2006).

aE. Mansfield. Algorithms for symmetric differential systems. Foundations of Computational Mathematics, 1:4 (2001).

aE. Mansfield. A Practical Guide to the Invariant Calculus. Cambridge Monographs on Applied and ComputationalMathematics 26. Cambridge University Press (2010).

aE. Hubert and I. A. Kogan. Rational invariants of a group action. Construction and rewriting. J. of Symbolic Computation,42:1-2 (2007).

aE. Hubert and I. A. Kogan. Smooth and algebraic invariants of a group action. Local and global constructions. Foundationsof Computational Mathematics, 7:4 (2007).

aE. Hubert. Differential invariants of a Lie group action: syzygies on a generating set. J. of Symbolic Computation , 44:3(2009).

aE. Hubert. Differential algebra for derivations with nontrivial commutation rules. J. of Pure and Applied Algebra, 200:1-2(2005).

aE. Hubert and P. J. Olver. Differential invariants of conformal and projective surfaces. Symmetry Integrability and Geometry:Methods and Applications, 3 (2007), Art. 097.

4 E. Hubert. Generation properties of Maurer-Cartan invariants. (Preliminary version) http://hal.inria.fr/inria-00194528/en.

`I. Anderson DifferentialGeometry . Maple 11-15.

`E. Hubert. diffalg: extension to non commuting derivations. INRIA, 2005. www.inria.fr/cafe/Evelyne.Hubert/diffalg.

O E. Hubert. aida - Algebraic Invariants and their Differential Algebra. INRIA, 2007. www.inria.fr/cafe/Evelyne.Hubert/aida.

1

1 Scaling symmetries

Dimensional analysisPrey-predator model [Murray, Mathematical Biology (2002)]{

n =((

1− nk1

)r − k2

pn+e

)n,

p = s(1−h pn

)p.

r, s, e, h, k1, k2 parameters.{n =

(1− n

k − h pn+1

)n,

p = s(1− p

n

)p.

s, h, k parameters

t → λ−1 t,n → µn,p → µν−1 p,

r → λ r,s → λ s,e → µ e,

h → ν h,k1 → µk1,k2 → λν k2

t = r t, n =n

e, p =

h p

e, s =

s

r, h =

k2

rh, k =

k1

e.

Symmetry reduction [Mansfield 01]

• Determine the scaling symmetry

• Compute a generating set of monomial invariants

• Rewrite the system in terms of those

It’s all linear algebra in the case of scalings!

Recipe for scaling reduction [H., Labahn 2012]

• Fill the matrix A determining the scaling

t → λ−1 t,n → µn,p → µν−1 p,

r → λ r,s → λ s,e → µ e,

h → ν h,k1 → µk1,k2 → λν k2

s r e h k1 k2 n p tν 0 0 0 1 0 1 0 −1 0µ 0 0 1 0 1 0 1 1 0λ 1 1 0 0 0 1 0 0 −1

A =

0 0 0 1 0 1 0 −1 00 0 1 0 1 0 1 1 01 1 0 0 0 1 0 0 −1

2

• Hermite Normal Form : there exists V ∈ Zn×n unimodular

AV = (H, 0)

A(Vi Vn

)=(H 0

)V −1 =

(Wu

Wd

)

V =

0 0 0 1 0 0 0 0 00 0 1 −1 0 −1 0 0 10 1 0 0 −1 0 −1 −1 01 0 0 0 0 −1 0 1 00 0 0 0 1 0 0 0 00 0 0 0 0 1 0 0 00 0 0 0 0 0 1 0 00 0 0 0 0 0 0 1 00 0 0 0 0 0 0 0 1

• Invariants:(

s r e h k1 k2 n p t)Vn =

(sr

k1e

k2r h

ne

h pe

r t)

(s r e h k1 k2 n p t

)Vn =(

s k h n p t)

• Rewriting: (s r e h k1 k2 n p t

)→(

s k h n p t)Wd

s→ s, r → 1, e→ 1, h→ 1, k1 → k, k2 → h, n→ n, p→ p, t→ t

A geometric picture

A =(

1 3)

A

(1 −30 1

)=(

1 0)

Scaling: X = λx, Y = λ3 y

Invariant: r = yx3

Rewriting: x→ 1, y → r

Symmetry reduction ' Projection on the cross-section along the orbits

3

2 Rational and algebraic invariants

Rational Action of an algebraic group G on M = KnG ⊂ Rl or Cl an algebraic variety G ⊂ K[λ1, . . . , λl] its ideal

K ⊂ C

g : G ×M → M(λ, z) 7→ λ ? z

(λ · µ) ? z = λ ? (µ ? z)

λ ? z =

(g1(λ, z)

h(λ, z), . . . ,

gn(λ, z)

h(λ, z)

)h, g1, . . . , gn ∈ K[λ1, . . . , λl, z1, . . . , zn]

Orbit of z ∈M: Oz = {λ ? z | λ ∈ G}

Rational Actions on R2

G R∗ SO(2)

G (λµ− 1) (λ2 + µ2 − 1)

λ ? z

(λxλ3y

) (λ −µµ λ

)(xy

)

Orbits

Field of Rational Invariants K(z)G

Rational invariant:p

q∈ K(z) p(λ?z)

q(λ?z) = p(z)q(z)

Field of rational invariants: K(z)G K(z)G = K(r1, . . . , rκ)

Separation: z′ ∈ Oz ⇔ r1(z′) = r1(z), . . . , rκ(z′) = rκ(z)for z, z′∈M \W

4

Algorithm [H., Kogan; JSC 07]

In : G, λ ? z , P

Out : {r1, . . . , rκ} ⊂ K(z)G

−→Q: K(z)G → K(y1, . . . , yκ)r 7→ R

r = R(r1, . . . , rκ)

So : K(z)G = K(r1, . . . , rκ)

Tool: Compute the Grobner basis Q of (P (Z) +G(λ) + (Z − λ ? z)) ∩K(z)[Z]A zero dimensional ideal.

Previous: [Rosenlicht 56], [Vinberg & Popov 89], [Muller-Quade & Beth 99]

Cross-section of degree eA variety P that intersects generic orbits in e simple points.

P defines a cross-section P of degree e:

• P ⊂ K[Z] prime ideal of complementary dimension to Oz

• I = (P +G+ (Z − λ ? z)) ∩K(z)[Z]radical and zero-dimensional

• dimK(z) K(z)[Z]/I = e

Take P = (ai1Z1 + . . .+ ainZn − bi, 1 ≤ i ≤ ro) or simply P =(Zi1 − b1, . . . , Ziro − bro

)

5

Examples

G K∗ SO(2)

λ ? z

(λxλ3y

) (λ −µµ λ

)(xy

)

O(X−λx, Y −λ3y

)(X−λx+µ y, Y −µx−λ y)

P +G (X − 1) + (λµ− 1) (X) + (λ2 + µ2 − 1)

Q X − 1, Y − yx3 X, Y 2 − (x2 + y2)

K(z)G r = yx3 r = x2 + y2

→ x→ 1, y → r x→ 0, y2 → r

Algebraic invariants [H. & Kogan, FoCM 07]

x → 0y2 → x2 + y2

(X,Y 2−(x2 + y2)

)z =

(0, ±

√x2 + y2

)

x → 0

y →√x2 + y2

Allowing algebraic invariants we could have the simplerrewrite rules

x→ 0, y →√x2 + y2

(P +G+ (Z − λ ? z)) ∩ R(z)G [Z] = (Q)

has e roots z = (z1, . . . , zn) in R(z)G .

Thm: In a neighborhood U of a point on P ,∃ z smooth; p1(z) = 0, . . . , pro(z) = 0

If f : U → R is a local invariant

f (z1, . . . , zn) = f (z1, . . . , zn) .

6

Geometric construction [Fels & Olver 99]

V1, . . . ,Vr the infinitesimal generators of the action

G R∗ SO(2)

λ ? z

(λxλ3y

) (λ −µµ λ

)(xy

)

V x ∂∂x + 3y ∂∂y y ∂∂x − x∂∂y

f : U → R smooth

f is a local invariant ⇔ V1(f) = 0, . . . ,Vr(f) = 0

Invariantization: ιf : U → Rthe only local invariant with ιf|P = f|P

ιf(z) = f(Z) where {Z} = Oz ∩ P

Normalized invariants

(ιz1, . . . , ιzn) = (z1, . . . , zn)

Prop: ιf(z1, . . . , zn) = f (ιz1, . . . , ιzn)

Symmetry reduction:(z1, . . . , zn)→ (ιz1, . . . , ιzn) subject to p1(ιz) = 0, . . . , pro(ιz) = 0

7

3 Differential Invariants

Classical differential invariants

E(2)(XY

)=

(λ −µµ λ

)(xy

)+

(αβ

)λ2 + µ2 = 1

YX =µ+ λyxλ− µ yx

YXX =yxx

(λ− µ yx)3

Curvature: σ =√

y2xx(1+y2x)3 a differential invariant

Invariant derivation:d

ds=

1√1 + y2

x

d

dx

Jets and Action Prolongation

J0 = X × U g(0) : G × J0 → J0 V01, . . . ,V

0r

(x1, . . . , xm) coordinates on X ; independent variables

(u1, . . . , un) coordinates on U ; dependent variables

Jk = X × U (k) g(k) : G × Jk → Jk Vk1 , . . . ,V

kr

additional coordinates uα =∂|α|u

∂xα11 . . . ∂xαmm

, |α| ≤ k

; the derivatives of u w.r.t x up to order k

Di =∂

∂xi+∑α

uα+εi

∂

∂uα

[DifferentialGeometry]

Differential Invariants

J0 = X × U g(0) : G × J0 → J0 V01, . . . ,V

0r

Jk = X × U (k) g(k) : G × Jk → Jk Vk1 , . . . ,V

kr

f : Jk → R differential invariant of order k if Vk(f) = 0.

Given a cross-section Pk on Jk, we define the normalized invariants of order k

Ik = {ιx1, . . . , ιxm} ∪ {ιuα | |α| ≤ k}

8

Invariant derivations [Fels & Olver 99]

D : F(Jk)→ F(Jk+1) s.t D ◦V = V ◦ D

f : Jk → R a differential invariant⇒ D(f) a differential invariant of order k + 1.

The dimensions rk of orbits on Jk stabilizes:r0 ≤ r1 ≤ . . . ≤ rs = rs+1 = . . . = r.

Vs1, . . . ,V

sr of rank r V(P ) = (Vi(pj)) full rank r

P = (p1, . . . , pr) a cross-section on Js+k

Ozz

ρ(z)

P

U

ρ : Jk → G equivariant ρ(λ ? z) = ρ(z) · λ−1

Invariant derivations: D1...Dm

= ρ∗ (Di(λ ? xj))ij

D1...

Dm

[Fels & Olver 99]

Di( ιf ) = ι ( Di(f) )−Kia ι (Va(f))

K = ι(D(P ) V(P )−1

)D(P ) = (Di(pj))ij

DiDj −DjDi =

m∑k=1

Λijk Dk

Λijk =

r∑c=1

Kic ι(Dj(ξck))−Kjc ι(Di(ξck))

9

Finite Generation [Fels & Olver 99] [Olver 07], [H. 09] [H. 12]

ιuα+εi = Di(ιuα) +Kia ι (Va(uα)) K = ι(D(P ) V(P )−1

)Any differential invariant can be contructively written in terms of:

• the normalized invariants of order s+ 1

Is+1 = {ιx1, . . . , ιxm} ∪ {ιuα | |α| ≤ s+ 1}

#Is+1 = m+ n

(s+ 1 +m

m

)• the edge invariants, when the cross-section is of minimal order

E = {ι(Di(pa))} ∪ I0

#E ≤ n+m(r + 1)

• the Maurer-Cartan invariants K = {Kia} ∪ I0

#K ≤ n+m(r+1)

and their derivatives w.r.t. D1, . . . ,Dm

Syzygies for Normalized Invariants [H. 09]

A subset S of the following relationships

p1(ιx, ιuα) = 0, . . . , pr(ιx, ιuα) = 0

Di(ιxj) = δij −Kiaι (V(xj)),

Di(ιuα) = ιuα+εi −Kiaι (V(uα)), |α| ≤ s

Di(ιuα)−Dj(ιuβ) = Kjaι (V(uβ))−Kiaι (V(uα)),

α+ εi = β + εj , |α| = |β| = s+ 1.

form a complete set of differential syzygies.

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ιuy

s + 1ιuxxιu ιux

ιuyy ιuα+εi = Di(ιuα) +Kia ι (Va(uα))

10

Maurer-Cartan inv. & Serret-Frenet [Mansfield & van der Kamp 06]

d

ds

tnbx

=

0 κ(s) 0 0

−κ(s) 0 τ(s) 00 −τ(s) 0 01 0 0 0

tnbx

E3 =

{(R 0at 1

)∣∣∣∣ R ∈ SO(3), a ∈ R3

}

κ

0 1 0 01 0 0 00 0 0 00 0 0 0

+ τ

0 0 0 00 0 −1 00 1 0 00 0 0 0

+ 1

0 0 0 00 0 0 00 0 0 01 0 0 0

+ 0

0 0 0 00 0 0 00 0 0 00 1 0 0

+ 0. . . .

K =(ιx2ss −

ιx3sss

ιx2ss

1 0 0)

=(κ τ 1 0 0

)

Syzygies on M-C Invariants [H. 2012]

P (Kia) = 0

Di(Kjc)−Dj(Kic) =∑

1≤a<b≤r

Cabc(KiaKjb −KjaKib) +

m∑k=1

ΛijkKkc = 0

where

[vi, vj ] =

r∑k=1

Cijkvk[Di,Dj

]=

m∑k=1

Λijk Dk.

11

4 Generalized Differential Algebra

Differential Polynomial Rings

F = Q(x, y)

∂1 =∂

∂x, ∂2 =

∂

∂y

Y = {φ, ψ}

F Jφ, ψK = F[φ, φx, φy, . . . , ψ . . .]

φxxy ; φx2y ; φ(2,1)

∂

∂x(φxxy) = φxxxy ; ∂1

(φ(2,1)

)= φ(3,1)

∂

∂x

∂

∂y=

∂

∂y

∂

∂x

F a field

∂ = {∂1, . . . , ∂m} derivations on F

Y = {y1, . . . , yn}

F[ yα | α ∈ Nm, y ∈ Y ] = F JYK

∂i (yα) = yα+εi

εi = (0, . . . , 1ith

, . . . , 0)

∂i∂j = ∂j∂i

Derivations with nontrivial commutations

Y = {y1, . . . , yn}

D = {D1, . . . ,Dm}

DiDj −Dj Di =

m∑l=1

ΛijlDl Λijl ∈ K JYK

K JYK?

Mononotone derivatives: yα = Dα11 . . .Dαmm y

Differential polynomial ring K JYK with non commuting derivations [H. 05]

Y = {y1, . . . , yn}

D = {D1, . . . ,Dm}

K[yα | α ∈ Nm, y ∈ Y]

Di (yα) =

yα+εiifα1 = . . . = αi−1 = 0

DjDi

(yα−εj

)+

m∑l=1

cijl Dl

(yα−εj

)where j < i is s.t. αj > 0

while α1 = . . . = αj−1 = 0

If the cijl satisfy

- cijl = −cjil

- Dk(cijl) + Di(cjkl) + Dj(ckil) =

m∑µ=1

cijµcµkl + cjkµcµil + ckiµcµjl

& there exists an admissible ranking ≺

- |α| < |β| ⇒ yα ≺ yβ ,

- yα ≺ zβ ⇒ yα+γ ≺ zβ+γ ,

-∑l∈Nm

cijlDl(yα) ≺ yα+εi+εj

then DiDj(p)−DjDi(p) =

m∑l=1

cijlDl(p)

∀p ∈ K[ yα |α ∈ Nm ] = K JYK

12

Results obtained with the software developed

[aida,diffalg]

Conformal & Projective Surfaces

A differential invariants for surfaces in R3 under the conformal/projective group canbe effectively written in terms of a one (two) differential invariants of order 3/4, andtheir invariant (monotone) derivatives. [H. & Olver 07]

Generalized Orthogonal Group O(3− l, l) nR3 acting on R3 × R:

Differential invariants of all orders can be effectively written in terms of 3 secondorder differential invariants. [H. 09]

13