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Voronoi's Algorithm ICERM Conference on Computational Challenges in the Theory of Lattices Providence, April 2018 Variations and Applications of Voronoi’s algorithm Achill Schürmann (Universität Rostock) ( based on work with Mathieu Dutour Sikiric and Frank Vallentin ) PRELUDE Voronoi’s Algorithm - classically - Lattices and Quadratic Forms Lattices and Quadratic Forms Lattices and Quadratic Forms Reduction Theory for positive definite quadratic forms n t GLn(Z) acts on by Q U QU S>0 7! Task of a reduction theory is to provide a fundamental domain Classical reductions were obtained by Lagrange, Gauß, Korkin and Zolotareff, Minkowski and others… All the same for n = 2 : Voronoi’s reduction idea Georgy Voronoi (1868 – 1908) Observation: The fundamental domain can be obtained from polyhedral cones that are spanned by rank-1 forms only Voronoi’s reduction idea Georgy Voronoi (1868 – 1908) Observation: The fundamental domain can be obtained from polyhedral cones that are spanned by rank-1 forms only Voronoi’s algorithm gives a recipe for the construction of a complete list of such polyhedral cones up to GL n ( Z ) -equivalence Perfect Forms Perfect Forms DEF: min(Q)= min Q[x] is the arithmetical minimum x n 0 2Z \{ } Perfect Forms DEF: min(Q)= min Q[x] is the arithmetical minimum x n 0 2Z \{ } Q is uniquely determined by min(Q) and n DEF: Q >0 perfect 2 S , n MinQ = x Z : Q[x]=min(Q) { 2 } Perfect Forms DEF: min(Q)= min Q[x] is the arithmetical minimum x n 0 2Z \{ } Q is uniquely determined by min(Q) and n DEF: Q >0 perfect 2 S , n MinQ = x Z : Q[x]=min(Q) { 2 } DEF: For Q n , its Voronoi cone is (Q)=cone xxt : x MinQ 2 S>0 V { 2 } Perfect Forms DEF: min(Q)= min Q[x] is the arithmetical minimum x n 0 2Z \{ } Q is uniquely determined by min(Q) and n DEF: Q >0 perfect 2 S , n MinQ = x Z : Q[x]=min(Q) { 2 } DEF: For Q n , its Voronoi cone is (Q)=cone xxt : x MinQ 2 S>0 V { 2 } THM: Voronoi cones give a polyhedral tessellation of n S>0 and there are only finitely many up to GL n ( Z ) -equivalence. Perfect Forms DEF: min(Q)= min Q[x] is the arithmetical minimum x n 0 2Z \{ } Q is uniquely determined by min(Q) and n DEF: Q >0 perfect 2 S , n MinQ = x Z : Q[x]=min(Q) { 2 } DEF: For Q n , its Voronoi cone is (Q)=cone xxt : x MinQ 2 S>0 V { 2 } THM: Voronoi cones give a polyhedral tessellation of n S>0 and there are only finitely many up to GL n ( Z ) -equivalence. (Voronoi cones are full dimensional if and only if Q is perfect!) Ryshkov Polyhedron The set of all positive definite quadratic forms / matrices with arithmetical minimum at least 1 is called Ryshkov polyhedron Ryshkov Polyhedron The set of all positive definite quadratic forms / matrices with arithmetical minimum at least 1 is called Ryshkov polyhedron = Q n : Q[x] 1 for all x Zn 0 R 2 S>0 ≥ 2 \{ } Ryshkov Polyhedron The set of all positive definite quadratic forms / matrices with arithmetical minimum at least 1 is called Ryshkov polyhedron Ryshkov Polyhedra = Q n : Q[x] 1 for all x Zn 0 R 2 S>0 ≥ 2 \{ } is a locallyis a finitelocally polyhedron finite polyhedron • R • R Vertices of are perfect forms • R 1 n n ↵ (det(Q + ↵Q0)) is strictly concave on • 7! S>0 Ryshkov Polyhedron The set of all positive definite quadratic forms / matrices with arithmetical minimum at least 1 is called Ryshkov polyhedron Ryshkov Polyhedra = Q n : Q[x] 1 for all x Zn 0 R 2 S>0 ≥ 2 \{ } is a locallyis a finitelocally polyhedron finite polyhedron • R • R • Vertices of are perfect Vertices of are perfectR forms • R 1 n n ↵ (det(Q + ↵Q0)) is strictly concave on • 7! S>0 Voronoi’sVoronoi’s algorithm Algorithm Start with a perfect form Q 1. SVP: Compute Min Q and describing inequalities of the polyhedral cone n (Q)= Q0 : Q0[x] 1 for all x Min Q P { 2 S ≥ 2 } 2. PolyRepConv: Enumerate extreme rays R ,...,R of (Q) 1 k P 3. SVPs: Determine contiguous perfect forms Qi = Q + ↵Ri, i =1,...,k 4. ISOMs: Test if Qi is arithmetically equivalent to a known form 5. Repeat steps 1.–4. for new perfect forms Computational Results Computational Results Computational Results 5000000 Wessel van Woerden, 2018 ?! Adjacency Decomposition Method (for vertex enumeration) Adjacency Decomposition Method (for vertex enumeration) • Find initial orbit(s) / representing vertice(s) Adjacency Decomposition Method (for vertex enumeration) • Find initial orbit(s) / representing vertice(s) • For each new orbit representative • enumerate neighboring vertices Adjacency Decomposition Method (for vertex enumeration) • Find initial orbit(s) / representing vertice(s) • For each new orbit representative • enumerate neighboring vertices • add as orbit representative if in a new orbit Adjacency Decomposition Method (for vertex enumeration) • Find initial orbit(s) / representing vertice(s) • For each new orbit representative • enumerate neighboring vertices (up to symmetry) • add as orbit representative if in a new orbit Representation conversion problem Adjacency Decomposition Method (for vertex enumeration) • Find initial orbit(s) / representing vertice(s) • For each new orbit representative • enumerate neighboring vertices (up to symmetry) • add as orbit representative if in a new orbit Representation conversion problem BOTTLENECK: Stabilizer and In-Orbit computations => Need of efficient data structures and algorithms for permutation groups: BSGS, (partition) backtracking Representation Conversion in practice Representation Conversion in practice Best known Algorithm: Mathieu Representation Conversion in practice Best known Algorithm: Mathieu A C++-Tool also available through polymake helps to compute linear automorphism groups • Thomas Rehn • converts polyhedral representations using (Phd 2014) Recursive Decomposition Methods (Incidence/Adjacency) http://www.geometrie.uni-rostock.de/software/ Applicaton: Lattice Sphere Packings The lattice sphere packing problem can be phrased as: Applicaton: Lattice Sphere Packings The lattice sphere packing problem can be phrased as: Part II: Koecher’s generalization and T-perfect forms Koecher’s generalization 1960/61 Max Koecher generalized Voronoi’s reduction theory and proofs to a setting with a self-dual cone C Max Koecher, 1924-1990 Koecher’s generalization 1960/61 Max Koecher generalized Voronoi’s reduction theory and proofs to a setting with a self-dual cone C Max Koecher, 1924-1990 Under certain conditions, he shows that C is covered by a tessellation of polyhedral Voronoi cones and “approximated from inside“ by a Ryshkov polyhedron Koecher’s generalization 1960/61 Max Koecher generalized Voronoi’s reduction theory and proofs to a setting with a self-dual cone C Max Koecher, 1924-1990 Under certain conditions, he shows that C is covered by a tessellation of polyhedral Voronoi cones and “approximated from inside“ by a Ryshkov polyhedron Can in particular be applied to obtain reduction domains for the action of GL ( ) on suitable quadratic spaces n OK Applications in Math Ryshkov Polyhedron ( ) symmetric O Applications in Math Ryshkov Polyhedron Vertices / Perfect Forms: • Reduction theory Hermite constant Representation • Conversion Polyhedral complex: • Cohomology of ( ) O Hecke operators ( ) symmetric • O • Compactifications of moduli spaces of Abelian varieties Applications in Math Ryshkov Polyhedron Vertices / Perfect Forms: • Reduction theory Hermite constant Representation • Conversion Polyhedral complex: • Cohomology of ( ) O Hecke operators ( ) symmetric • O • Compactifications of See Mathieu’s talk moduli spaces after the coffee break! of Abelian varieties AIM Square group 2012: Gangl, Dutour Sikirić, Schürmann, Gunnells, Yasaki, Hanke Embedding Koecher’s theory For practical computations: Koecher’s theory can be embedded into a linear subspace T in some higher dimensional space of symmetric matrices EmbeddingEquivariant theory Koecher’s theory For a finite group G GLn(Z) the space of invariant forms For practical computations:⇢ Koecher’s theory can be embedded T := Q n : G Aut Q G into a {linear2 S subspace⇢ T } is ain linear some subspace higher of dimensionaln; T spacen is calledof symmetricBravais space matrices S G \ S>0 IDEA (Berge,´ Martinet, Sigrist, 1992): Intersect Ryshkov polyhedron with a linear subspace T n R ⇢ S EmbeddingEquivariant theory Koecher’s theory For a finite group G GLn(Z) the space of invariant forms For practical computations:⇢ Koecher’s theory can be embedded T := Q n : G Aut Q G into a {linear2 S subspace⇢ T } is ain linear some subspace higher of dimensionaln; T spacen is calledof symmetricBravais space matrices S G \ S>0 IDEA (Berge,´ Martinet, Sigrist, 1992): Intersect Ryshkov polyhedron with a linear subspace T n R ⇢ S T-perfect and T-extreme forms DEF: Q T n T-perfect and T-extreme2 \ S>0 forms is T -extreme if it attains a loc. max. of δ within T • DEF: Q T n is T -perfect if it is a vertex of T 2 \ S>•0 R \ 1 is T -extreme if it attainsis T -eutactic a loc. max.if Q of− δ withinT relint(T (Q) T ) • • | 2 V | is T -perfect if it is a vertex of T • R \ 1 is T -eutacticTHMif Q− (BMS,T 1992):relint( (QTQ) -extremeT ) QT-perfect and T -eutactic • | 2 V | , THM (BMS, 1992): QT-extreme QT-perfect and T -eutactic ,n Q, Q0 T >0 are called T -equivalent, if U GLn(Z) with • 2 \ S 9 2 t t 0 n Q = U QU and T = U TU Q, Q0 T >0 are called T -equivalent, if U GLn(Z) with • 2 \ S 9 2 t t Q0 = U QU and T = U TU THM (Jaquet-Chiffelle, 1995): T -perfect Q : λ(Q)=1 / finite { G } ⇠TG THM (Jaquet-Chiffelle, 1995): TVoronoi’s-perfect Q algorithm: λ(Q)=1 can be/ appliedfinite to TG ){ G } ⇠TG R \ Voronoi’s algorithm can be applied to T ) R \ G T-Algorithm Voronoi’s Algorithm for a linear subspace T SVPs: Obtain a T -perfect form Q 1.
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