Multi-parameter persistent homology: applications and algorithms

Transcription

1 Multi-parameter persistent homology: applications and algorithms Nina Otter Mathematical Institute, University of Oxford Gudhi/Top Data Workshop Porquerolles, 18 October 2016

2 Multi-parameter persistent homology pipeline Data depending on r parameters (1) Multi-filtered (2) space Multiparameter persistence module

3 Step (1): from data to multi-filtered spaces Define the following partial order on N r : (u 1,..., u r ) (v 1,..., v r ) iff u i v i for all i = 1,..., r. A multi-filtered space K is a set of spaces {K u } u N r such that K u K v if u v for all u, v N r.

4 Step (1): from data to multi-filtered spaces Define the following partial order on N r : (u 1,..., u r ) (v 1,..., v r ) iff u i v i for all i = 1,..., r. A multi-filtered space K is a set of spaces {K u } u N r such that K u K v if u v for all u, v N r. Map f : X R r r-filtered simplicial complex digital image with color vectors of length r r-filtered cubical complex

5 Step (1): from data to multi-filtered spaces: example G R

6 Step (2): from multi-filtered spaces to multi-parameter persistence modules r-filtered space H i r-parameter persistence module An r-parameter persistence module is a tuple ({M i } i N r, {φ i,j } i j N r ) where: for each i N r we have that M i is a k-module for every i j we have that φ i,j : M i M j is a k-module homomorphism such that whenever i k j we have φ k,j φ i,k = φ i,j. In other words, an r-parameter persistence module is a functor F : N r kmod.

7 Interlude: representation theory of quivers

8 Interlude: representation theory of quivers A quiver Q = (Q 0, Q 1, s, t) consists of two non-empty sets Q 0, Q 1 and two maps s, t : Q 1 Q 0. A quiver is finite if both Q 0 and Q 1 are finite. Whenever s(u) = x and t(u) = y we write x the following are finite quivers: u y. For example, s u x v y t w h

9 Representations of quivers Let k be a field. A representation of a quiver (V, φ) consists of a family of k-vector spaces V = {V i } i Q0 together with a family of k-linear maps φ = {φ e : V s(e) Vt(e) e Q 1 }. A representation (V, φ) is finite-dimensional if for all i Q 0 the vector space V i is finite-dimensional. A morphism of representations f : (V, φ) (V, φ ) is given by k-linear maps f i : V i V i for all i Q 0 such that the following diagram V s(e) φ e V t(e) f s(e) f t(e) V s(e) V t(e) φ t(e) commutes for all e Q 1.

10 Examples of quiver representations x u y u x u x v Two finite-dimensional representations φ: V V and ψ : W W are isomorphic iff dim V = dim W and dim V = dim W and rankφ = rankψ. Two finite-dimensional representations φ: V V and ψ : W W are isomorphic iff φ and ψ have the same Jordan normal form. Studying isomorphism classes of representations of this quiver amounts to studying pairs of quadratic matrices up to simultaneous conjugation.

11 Indecomposable representations The direct sum of two representations (φ, V ) and (ψ, W ) is the representation (φ ψ, V ( W ) where ) V W = V i W i for all φe 0 i Q 0 and (φ ψ) e =. 0 ψ e

12 Indecomposable representations The direct sum of two representations (φ, V ) and (ψ, W ) is the representation (φ ψ, V ( W ) where ) V W = V i W i for all φe 0 i Q 0 and (φ ψ) e =. 0 ψ e We say that a representation (φ, V ) is indecomposable if it is non-zero and not isomorphic to a direct sum of two non-zero representations.

13 Indecomposable representations The direct sum of two representations (φ, V ) and (ψ, W ) is the representation (φ ψ, V ( W ) where ) V W = V i W i for all φe 0 i Q 0 and (φ ψ) e =. 0 ψ e We say that a representation (φ, V ) is indecomposable if it is non-zero and not isomorphic to a direct sum of two non-zero representations. Example: indecomposable representations of the loop quiver are given by the Jordan blocks.

14 Indecomposable representations The direct sum of two representations (φ, V ) and (ψ, W ) is the representation (φ ψ, V ( W ) where ) V W = V i W i for all φe 0 i Q 0 and (φ ψ) e =. 0 ψ e We say that a representation (φ, V ) is indecomposable if it is non-zero and not isomorphic to a direct sum of two non-zero representations. Example: indecomposable representations of the loop quiver are given by the Jordan blocks. Theorem (Krull, Remak, Schmidt) Assume that Q is finite, then any finite-dimensional representation (V, φ) of Q can be written as a direct sum (V, φ) = (V 1, φ 1 ) (V r, φ r ) where each (V i, φ i ) is indecomposable, and the decomposition is unique up to isomorphism and permutation of the terms.

15 Classification of (representations of) quivers Dynkin (n 2) Extended Dynkin (n 2) Wild n 1 n Everything else. For example: n 1 n n 1 n 1 n n...

16 Classification of representations of quivers Suppose that k is algebraically closed. The number of isomorphism classes of indecomposable representations is: Dynkin Finite. Extended Dynkin Infinite; depends on one parameter. Wild Infinite; depends on N > 1 parameters, where N depends on the quiver. G. Kac, Infinite root systems, representations of graphs and invariant theory I - II, and P. Gabriel, Unzerlegbare darstellungen I, 1972.

17 Classification of indecomposable representations of quivers: example Conside again the loop quiver:

18 Classification of indecomposable representations of quivers: example Conside again the loop quiver: Recall that two finite-dimensional representations φ: V V and ψ : W W are isomorphic iff φ and ψ have the same Jordan normal form, and the isomorphism classes of indecomposable representations of the loop quiver are given by the Jordan blocks. Each Jordan block depends on a continuous parameter given by the eigenvalue.

19 Back to multi-parameter persistent homology A multi-parameter persistence module is a representation of a quiver of the following form:

20 Back to multi-parameter persistent homology A multi-parameter persistence module is a representation of a quiver of the following form: Such quivers are wild.

21 Back to multi-parameter persistent homology A multi-parameter persistence module is a representation of a quiver of the following form: Such quivers are wild. Our motivation/goal: find computable invariants for applications.

22 Application: Time evolution of blood vessel growth in presence of tumors Roche, Oxford Oncology (B. Markelc), Mathematical Biology, University of Oxford (B. Stolz, H. Byrne, J. Grogan)

23 Persistence modules are modules

24 Persistence modules are modules Recall that an N r -graded (or multi-graded) ring is a ring A together with a collection {A u } u N r of subgroups of the underlying abelian group of A such that A = u N r A u and for all a A m and b A n we have ab A m+n.

25 Persistence modules are modules Recall that an N r -graded (or multi-graded) ring is a ring A together with a collection {A u } u N r of subgroups of the underlying abelian group of A such that A = u N r A u and for all a A m and b A n we have ab A m+n. Make the ring A = k[x 1,..., x r ] into an N r -graded ring by setting A u = kx u x ur r for all u = (u 1,..., u r ) N r.

26 Persistence modules are modules Recall that an N r -graded (or multi-graded) ring is a ring A together with a collection {A u } u N r of subgroups of the underlying abelian group of A such that A = u N r A u and for all a A m and b A n we have ab A m+n. Make the ring A = k[x 1,..., x r ] into an N r -graded ring by setting A u = kx u x ur r for all u = (u 1,..., u r ) N r. A module M over an N r -graded ring A is graded if there is a collection {M i } i N r of subgroups of the underlying abelian group of M such that M = i N r M i and for all a A j we have am i M i+j.

27 Persistence modules are modules Recall that an N r -graded (or multi-graded) ring is a ring A together with a collection {A u } u N r of subgroups of the underlying abelian group of A such that A = u N r A u and for all a A m and b A n we have ab A m+n. Make the ring A = k[x 1,..., x r ] into an N r -graded ring by setting A u = kx u x ur r for all u = (u 1,..., u r ) N r. A module M over an N r -graded ring A is graded if there is a collection {M i } i N r of subgroups of the underlying abelian group of M such that M = i N r M i and for all a A j we have am i M i+j. Correspondence Theorem of Persistent Homology (Carlsson, Zomorodian 09) The functor category of r-parameter persistence modules is isomorphic to the category of graded k[x 1,..., x r ]-modules and module homomorphisms respecting the grading.

28 Any persistence module is the homology of a filtered space The homology of a multi-filtered space is a persistence module.

29 Any persistence module is the homology of a filtered space The homology of a multi-filtered space is a persistence module. On the other hand: Theorem (Carlsson, Zomorodian, 2009) For any finite persistence module M there exists a multi-filtered space K and a positive natural number i such that M is the homology in degree i of K.

30 Any persistence module is the homology of a filtered space The homology of a multi-filtered space is a persistence module. On the other hand: Theorem (Carlsson, Zomorodian, 2009) For any finite persistence module M there exists a multi-filtered space K and a positive natural number i such that M is the homology in degree i of K. Therefore, studying the homology of r-filtered spaces amounts to studying graded modules over k[x 1,..., x r ].

31 Free resolutions and presentations Let M be a finitely generated graded k[x 1,..., x r ]-module. By the Hilbert Syzygy Theorem there is a free resolution by finitely generated N r -graded free k[x 1,..., x r ]-modules of length at most r: φ m φ 1 0 F m Fm 1... F 1 F0 M 0 with image(φ i ) = kernel(φ i 1 ) and each F i is a finitely generated graded free k[x 1,..., x r ]-module and m r.

32 Free resolutions and presentations Let M be a finitely generated graded k[x 1,..., x r ]-module. By the Hilbert Syzygy Theorem there is a free resolution by finitely generated N r -graded free k[x 1,..., x r ]-modules of length at most r: φ m φ 1 0 F m Fm 1... F 1 F0 M 0 with image(φ i ) = kernel(φ i 1 ) and each F i is a finitely generated graded free k[x 1,..., x r ]-module and m r. The first part φ 1 F 1 F0 M 0 of a free resolution of a module is called presentation. If we are given a presentation of M, we can then explicitly write M as the quotient F 0 /imφ 1.

33 Minimal presentations and resolutions Resolutions and presentations are in general not unique.

34 Minimal presentations and resolutions Resolutions and presentations are in general not unique. Example: 1 Let M = (x 1 x 2, x 1 x 3 ) k[x 1, x 2, x 3 ] = S. The following are two free resolutions of M: x 3 ( ) x 2 x1 x 0 S S 2 2 x 1 x 3 M 0 x 2 x 3 x 2 x 3 1 x 0 S 2 2 x2 2 S 2 ( x1 x 2 x 1 x 3 ) M 0 However, minimal presentations of modules over local or graded rings are unique up to isomorphism. 1 Bulletin of the AMS, July 2016

35 Invariants from resolutions and presentations Minimal presentations are invariants of a module, and one can compute many invariants from minimal presentations and resolutions, such as: Betti numbers (Multi-graded) Hilbert series...

36 Presentation of a persistence module: naïve Algorithm Since the ith homology of the ith chain complex of a multi-filtered simplicial complex is defined as H i = kernel(d i )/image(d i+1 ), an algorithm to compute a presentation of H i is given by the following steps: 1. Compute a presentation of image(d i+1 ). 1 G. Carlsson, G. Singh, A. Zomorodian, Computing multidimensional persistence, 2010.

37 Presentation of a persistence module: naïve Algorithm Since the ith homology of the ith chain complex of a multi-filtered simplicial complex is defined as H i = kernel(d i )/image(d i+1 ), an algorithm to compute a presentation of H i is given by the following steps: 1. Compute a presentation of image(d i+1 ). 2. Compute a presentation of kernel(d i ). 1 G. Carlsson, G. Singh, A. Zomorodian, Computing multidimensional persistence, 2010.

38 Presentation of a persistence module: naïve Algorithm Since the ith homology of the ith chain complex of a multi-filtered simplicial complex is defined as H i = kernel(d i )/image(d i+1 ), an algorithm to compute a presentation of H i is given by the following steps: 1. Compute a presentation of image(d i+1 ). 2. Compute a presentation of kernel(d i ). 3. Compute a presentation of the quotient H i. Problem: the known algorithms to compute image(d i+1 ) are exponential in time and space 1. 1 G. Carlsson, G. Singh, A. Zomorodian, Computing multidimensional persistence, 2010.

39 Presentation of a module: algorithm by Carlsson, Singh and Zomorodian In 2010 Carlsson, Singh and Zomorodian 2 put forward an optimisation of the algorithm to compute a presentation of homology of one-critical multi-filtered complexes. 2 G. Carlsson, G. Singh, and A. Zomorodian, Computing multidimensional persistence, 2010.

40 Presentation of a module: algorithm by Carlsson, Singh and Zomorodian In 2010 Carlsson, Singh and Zomorodian 2 put forward an optimisation of the algorithm to compute a presentation of homology of one-critical multi-filtered complexes. Let K = {K u } u N r be a multi-filtered simplicial complex. We assume that there exists v N r such that K v is a finite simplicial complex, and K u = K v for all u v. We denote the simplicial complex K v by K tot. 2 G. Carlsson, G. Singh, and A. Zomorodian, Computing multidimensional persistence, 2010.

41 Presentation of a module: algorithm by Carlsson, Singh and Zomorodian In 2010 Carlsson, Singh and Zomorodian 2 put forward an optimisation of the algorithm to compute a presentation of homology of one-critical multi-filtered complexes. Let K = {K u } u N r be a multi-filtered simplicial complex. We assume that there exists v N r such that K v is a finite simplicial complex, and K u = K v for all u v. We denote the simplicial complex K v by K tot. For any σ K tot define the set of generators of σ to be gen(σ) = min{v N r σ K(v)}. 2 G. Carlsson, G. Singh, and A. Zomorodian, Computing multidimensional persistence, 2010.

42 Presentation of a module: algorithm by Carlsson, Singh and Zomorodian In 2010 Carlsson, Singh and Zomorodian 2 put forward an optimisation of the algorithm to compute a presentation of homology of one-critical multi-filtered complexes. Let K = {K u } u N r be a multi-filtered simplicial complex. We assume that there exists v N r such that K v is a finite simplicial complex, and K u = K v for all u v. We denote the simplicial complex K v by K tot. For any σ K tot define the set of generators of σ to be gen(σ) = min{v N r σ K(v)}. A multi-filtered simplicial complex is one-critical if the set of generators of every simplex has cardinality one. 2 G. Carlsson, G. Singh, and A. Zomorodian, Computing multidimensional persistence, 2010.

43 One-critical multi-filtered simplicial complex: example One-critical: (0, 1) (1, 1) (0, 0) (1, 0)

44 One-critical multi-filtered simplicial complex: example One-critical: Not one-critical: (0, 1) (1, 1) (0, 0) (1, 0) (0, 1) (1, 1) (1, 0)

45 Optimization For a one-critical multifiltered simplicial complex K: the chain modules are free modules, hence one can choose bases for them The standard basis is the basis of simplices in degree given by their generator The boundary maps can be written as homogeneous matrices with monomial entries Carlsson, Singh and Zomorodian show that this gives a polynomial bound on complexity. The resulting presentation is not an invariant, as it depends on a choice of basis.

46 Presentation of a module: algorithm by Chacholski-Scolamiero-Vaccarino (CSV) In 2014 Chacholski, Scolamiero and Vaccarino 3 put forward a polynomial-time algorithm to compute a presentation of homology of arbitrary multi-filtered simplicial complexes. 3 W. Chacholski, M. Scolamiero, and F. Vaccarino, Combinatorial presentation of multidimensional persistent homology, 2014.

47 Presentation of a module: algorithm by Chacholski-Scolamiero-Vaccarino (CSV) In 2014 Chacholski, Scolamiero and Vaccarino 3 put forward a polynomial-time algorithm to compute a presentation of homology of arbitrary multi-filtered simplicial complexes. For any u N r, denote by K n,u the set of n-simplices in K u ; the assignment u K n,u induces a functor K n : N r Sets, where Sets is the category of sets. For any v N r and any i {0,..., n + 1} define the following map d i : K n+1,v K n,v : {x 0,..., x n+1 } {x 0,..., ˆx i,..., x n+1 } where ˆx i means that we omit the vertex x i. The maps d i give natural transformations K n+1 K n. 3 W. Chacholski, M. Scolamiero, and F. Vaccarino, Combinatorial presentation of multidimensional persistent homology, 2014.

48 CSV algorithm Let S = k[x 1,..., x r ].There exists a sequence of free graded S-modules RKK n RGK n+1 π d α RGK n RD n 1 such that α (π d) is trivial, and the k[x 1,..., x r ]-module kernel(α)/im(π d) is isomorphic to the homology in degree n of the multi-filtered simplicial complex K,

49 CSV algorithm Let S = k[x 1,..., x r ].There exists a sequence of free graded S-modules RKK n RGK n+1 π d α RGK n RD n 1 such that α (π d) is trivial, and the k[x 1,..., x r ]-module kernel(α)/im(π d) is isomorphic to the homology in degree n of the multi-filtered simplicial complex K, where RKK n = RGK n = RD n 1 = σ K tot,n v 0 v 1 gen(σ) σ K tot,n v gen(σ) σ K tot,n 1 S. x max{v 0,v 1 } S, x v S, and with the notation x u := x u x ur r.

50 CSV algorithm The homomorphisms are defined as follows: (π) For any σ K n and v 0 v 1 gen(σ), the homomorphism π : RKK n RGK n sends x max{v 0,v 1 } to x v 0 x v 1. (d) For any σ K n+1 and v gen(σ), the homomorphism d : RGK n+1 RGK n sends x v to n+1 i=0 ( 1)i x d i (σ), where d i (σ) is the minimal element in the set {w gen(d i (σ)) w v} with respect to the lexicographical order. (α) For any σ K n and v gen(σ), the homomorphism α: RGK n RD n 1 sends x v to n i=0 ( 1)i d i (σ).

51 CSV algorithm: example We illustrate the algorithm for the computation of H 1 of the following 2-filtered simplicial complex K: c c (0, 1) a b (1, 1) a b c (1, 0) a b

52 CSV algorithm: example We illustrate the algorithm for the computation of H 1 of the following 2-filtered simplicial complex K: c c We have: (0, 1) a b (1, 1) a b gen({a}) = {(0, 1), (1, 0)} c (1, 0) a b

53 CSV algorithm: example We illustrate the algorithm for the computation of H 1 of the following 2-filtered simplicial complex K: c c We have: (0, 1) a b (1, 1) a b gen({a}) = {(0, 1), (1, 0)} gen({b}) = {(0, 1), (1, 0)} c (1, 0) a b

54 CSV algorithm: example We illustrate the algorithm for the computation of H 1 of the following 2-filtered simplicial complex K: c c We have: (0, 1) a b (1, 1) a c b gen({a}) = {(0, 1), (1, 0)} gen({b}) = {(0, 1), (1, 0)} gen({c}) = {(0, 1), (1, 0)} (1, 0) a b

55 CSV algorithm: example We illustrate the algorithm for the computation of H 1 of the following 2-filtered simplicial complex K: c c We have: (0, 1) a b (1, 1) a c (1, 0) a b b gen({a}) = {(0, 1), (1, 0)} gen({b}) = {(0, 1), (1, 0)} gen({c}) = {(0, 1), (1, 0)} gen({a, b}) = {(0, 1), (1, 0)}

56 CSV algorithm: example We illustrate the algorithm for the computation of H 1 of the following 2-filtered simplicial complex K: c c We have: (0, 1) a b (1, 1) a c (1, 0) a b b gen({a}) = {(0, 1), (1, 0)} gen({b}) = {(0, 1), (1, 0)} gen({c}) = {(0, 1), (1, 0)} gen({a, b}) = {(0, 1), (1, 0)} gen({b, c}) = {(0, 1), (1, 0)}

57 CSV algorithm: example We illustrate the algorithm for the computation of H 1 of the following 2-filtered simplicial complex K: c c We have: (0, 1) a b (1, 1) a c (1, 0) a b b gen({a}) = {(0, 1), (1, 0)} gen({b}) = {(0, 1), (1, 0)} gen({c}) = {(0, 1), (1, 0)} gen({a, b}) = {(0, 1), (1, 0)} gen({b, c}) = {(0, 1), (1, 0)} gen({a, c}) = {(0, 1), (1, 0)}

58 CSV algorithm: example We illustrate the algorithm for the computation of H 1 of the following 2-filtered simplicial complex K: c c We have: (0, 1) a b (1, 1) a c (1, 0) a b b gen({a}) = {(0, 1), (1, 0)} gen({b}) = {(0, 1), (1, 0)} gen({c}) = {(0, 1), (1, 0)} gen({a, b}) = {(0, 1), (1, 0)} gen({b, c}) = {(0, 1), (1, 0)} gen({a, c}) = {(0, 1), (1, 0)} gen({a, b, c}) = {(1, 1)}

59 CSV algorithm: example Let S = k[x 1, x 2 ]. Then: RKK 1 = σ K tot,1 v 0 v 1 gen(σ) x max{v 0,v 1 } S

60 CSV algorithm: example Let S = k[x 1, x 2 ]. Then: RKK 1 = σ K tot,1 v 0 v 1 gen(σ) x max{v 0,v 1 } S, so RKK 1 = x 1 x 2 S x 1 x 2 S x 1 x 2 S

61 CSV algorithm: example Let S = k[x 1, x 2 ]. Then: RKK 1 = σ K tot,1 v 0 v 1 gen(σ) x max{v 0,v 1 } S, so RKK 1 = x 1 x 2 S x 1 x 2 S x 1 x 2 S RGK n = σ K tot,n v gen(σ) x v S

62 CSV algorithm: example Let S = k[x 1, x 2 ]. Then: RKK 1 = σ K tot,1 v 0 v 1 gen(σ) x max{v 0,v 1 } S, so RKK 1 = x 1 x 2 S x 1 x 2 S x 1 x 2 S RGK n = σ K tot,n v gen(σ) x v S, so RGK 2 = x 1 x 2 S

63 CSV algorithm: example Let S = k[x 1, x 2 ]. Then: RKK 1 = σ K tot,1 v 0 v 1 gen(σ) x max{v 0,v 1 } S, so RKK 1 = x 1 x 2 S x 1 x 2 S x 1 x 2 S RGK n = σ K tot,n v gen(σ) x v S, so RGK 2 = x 1 x 2 S RGK 1 = x 1 S x 2 S x 1 S x 2 S x 1 S x 2 S

64 CSV algorithm: example Let S = k[x 1, x 2 ]. Then: RKK 1 = σ K tot,1 v 0 v 1 gen(σ) x max{v 0,v 1 } S, so RKK 1 = x 1 x 2 S x 1 x 2 S x 1 x 2 S RGK n = σ K tot,n v gen(σ) x v S, so RGK 2 = x 1 x 2 S RGK 1 = x 1 S x 2 S x 1 S x 2 S x 1 S x 2 S RD 0 = σ K tot,0 k[x 1,..., x r ], so

65 CSV algorithm: example Let S = k[x 1, x 2 ]. Then: RKK 1 = σ K tot,1 v 0 v 1 gen(σ) x max{v 0,v 1 } S, so RKK 1 = x 1 x 2 S x 1 x 2 S x 1 x 2 S RGK n = σ K tot,n v gen(σ) x v S, so RGK 2 = x 1 x 2 S RGK 1 = x 1 S x 2 S x 1 S x 2 S x 1 S x 2 S RD 0 = σ K tot,0 k[x 1,..., x r ], so RDK 0 = S S S

66 CSV algorithm: example π : RKK 1 RGK 1 sends x max{v 0,v 1 } to x v 0 x v 1, so

67 CSV algorithm: example π : RKK 1 RGK 1 sends x max{v 0,v 1 } to x v 0 x v 1, so x x π = 0 x x x x 2

68 CSV algorithm: example π : RKK 1 RGK 1 sends x max{v 0,v 1 } to x v 0 x v 1, so x x π = 0 x x x x 2 d : RGK 2 RGK 1 sends x v to n+1 i=0 ( 1)i x d i (σ)

69 CSV algorithm: example π : RKK 1 RGK 1 sends x max{v 0,v 1 } to x v 0 x v 1, so x x π = 0 x x x x 2 d : RGK 2 RGK 1 sends x v to n+1 i=0 ( 1)i x d i (σ), so d = ( 0 x 2 0 x 2 0 x 2 ) t

70 CSV algorithm: example π : RKK 1 RGK 1 sends x max{v 0,v 1 } to x v 0 x v 1, so x x π = 0 x x x x 2 d : RGK 2 RGK 1 sends x v to n+1 i=0 ( 1)i x d i (σ), so d = ( 0 x 2 0 x 2 0 x 2 ) t α: RGK 1 RD 0 sends x v to n i=0 ( 1)i d i (σ)

71 CSV algorithm: example π : RKK 1 RGK 1 sends x max{v 0,v 1 } to x v 0 x v 1, so x x π = 0 x x x x 2 d : RGK 2 RGK 1 sends x v to n+1 i=0 ( 1)i x d i (σ), so d = ( 0 x 2 0 x 2 0 x 2 ) t α: RGK 1 RD 0 sends x v to n i=0 ( 1)i d i (σ), so α =

72 CSV algorithm: example By using a computational algebra software package one can then compute the following minimal presentation: and thus x 1 0 ( ) 0 x 0 S 2 2 x2 x S 2 1 H 1 (K) 0 H 1 (K) = x 1S (x 1 x 2 ) x 2S (x 1 x 2 ).

73 CSV algorithm: example H 1 (K) = x 1S (x 1 x 2 ) x 2S (x 1 x 2 ).

74 Conclusions

75 Conclusions Need efficient implementation of algorithm by Chacholski, Scolamiero and Vaccarino. Computational algebra libraries are not efficient. How complex is the problem in practice? Insight from geometric invariant theory?