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Background and problem statement Using Gurvits algorithm Progress via Blow-ups Null cone membership for the left right action on tuples of matrices Gabor Ivanyos 1 , Jimmy Qiao 2 , K V Subrahmanyam 3 1 Institute for Computer Science and Control,

  1. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Invariant theory Membership in the null cone Null cone for the left right action Is defined as the m -tuple of n by n matrices on which all all invariant polynomial functions vanish i.e f p B 1 , B 2 , . . . , B m q “ 0 for all invariant polynomial functions f . Over infinite fields - an alternate characterization - ( B 1 , B 2 , . . . , B m ) such that B has a c -shrunk subspace for c ą 0 [BD06, DZ01, ANS07].

  2. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Invariant theory Membership in the null cone Null cone for the left right action Is defined as the m -tuple of n by n matrices on which all all invariant polynomial functions vanish i.e f p B 1 , B 2 , . . . , B m q “ 0 for all invariant polynomial functions f . Over infinite fields - an alternate characterization - ( B 1 , B 2 , . . . , B m ) such that B has a c -shrunk subspace for c ą 0 [BD06, DZ01, ANS07]. A description of the invariants: Let T 1 , T 2 , . . . , T m be matrices in M at p d , F q . Then det p T 1 b X 1 ` T 2 b X 2 ` . . . ` T m b X m q is an invariant of degree nd .

  3. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Invariant theory Membership in the null cone Null cone for the left right action Is defined as the m -tuple of n by n matrices on which all all invariant polynomial functions vanish i.e f p B 1 , B 2 , . . . , B m q “ 0 for all invariant polynomial functions f . Over infinite fields - an alternate characterization - ( B 1 , B 2 , . . . , B m ) such that B has a c -shrunk subspace for c ą 0 [BD06, DZ01, ANS07]. A description of the invariants: Let T 1 , T 2 , . . . , T m be matrices in M at p d , F q . Then det p T 1 b X 1 ` T 2 b X 2 ` . . . ` T m b X m q is an invariant of degree nd . Over infinite fields, all invariants are obtained this way.

  4. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Outline 1 Background and problem statement Problem statement Invariant theory 2 Using Gurvits algorithm 3 Progress via Blow-ups Regularity Algorithmic and degree bounds Degree bounds Polynomial bound - degree of generation Main lemma and blow ups using division algebras Proof of the main lemma Matrix of maximum rank

  5. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Suggested algorithm

  6. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Suggested algorithm Observation If B 1 shrinks a subspace U P F n , and T 1 P M at p d , F q then T 1 b B 1 shrinks the subspace U b F d .

  7. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Suggested algorithm Observation If B 1 shrinks a subspace U P F n , and T 1 P M at p d , F q then T 1 b B 1 shrinks the subspace U b F d . If B shrinks U , then so will its d -th blow-up

  8. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Suggested algorithm Observation If B 1 shrinks a subspace U P F n , and T 1 P M at p d , F q then T 1 b B 1 shrinks the subspace U b F d . If B shrinks U , then so will its d -th blow-up B t d , d u := x T 1 b B 1 , T 2 b B 2 , . . . , T m b B m y , T i P M at p d , F q .

  9. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Suggested algorithm Observation If B 1 shrinks a subspace U P F n , and T 1 P M at p d , F q then T 1 b B 1 shrinks the subspace U b F d . If B shrinks U , then so will its d -th blow-up B t d , d u := x T 1 b B 1 , T 2 b B 2 , . . . , T m b B m y , T i P M at p d , F q . for i “ 1 , 2 , . . . , compute (a basis of) B t i , i u .

  10. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Suggested algorithm Observation If B 1 shrinks a subspace U P F n , and T 1 P M at p d , F q then T 1 b B 1 shrinks the subspace U b F d . If B shrinks U , then so will its d -th blow-up B t d , d u := x T 1 b B 1 , T 2 b B 2 , . . . , T m b B m y , T i P M at p d , F q . for i “ 1 , 2 , . . . , compute (a basis of) B t i , i u . determine if there is a nonsingular matrix in the blow-up.

  11. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Suggested algorithm Observation If B 1 shrinks a subspace U P F n , and T 1 P M at p d , F q then T 1 b B 1 shrinks the subspace U b F d . If B shrinks U , then so will its d -th blow-up B t d , d u := x T 1 b B 1 , T 2 b B 2 , . . . , T m b B m y , T i P M at p d , F q . for i “ 1 , 2 , . . . , compute (a basis of) B t i , i u . determine if there is a nonsingular matrix in the blow-up. Question How long do we go on?

  12. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Implications of degree bound σ Theorem [IQS15a] Over Q , if the nullcone is defined by elements of degree ď σ “ σ p n , m q ,there exists a deterministic poly( n , m , σ ) algorithm deciding if ( B 1 , B 2 , . . . , B m ) is in the null cone.

  13. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Implications of degree bound σ Theorem [IQS15a] Over Q , if the nullcone is defined by elements of degree ď σ “ σ p n , m q ,there exists a deterministic poly( n , m , σ ) algorithm deciding if ( B 1 , B 2 , . . . , B m ) is in the null cone. If p B 1 , . . . , B m q is in the null cone all blow-ups B t d , d u shrink a subspace.

  14. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Implications of degree bound σ Theorem [IQS15a] Over Q , if the nullcone is defined by elements of degree ď σ “ σ p n , m q ,there exists a deterministic poly( n , m , σ ) algorithm deciding if ( B 1 , B 2 , . . . , B m ) is in the null cone. If p B 1 , . . . , B m q is in the null cone all blow-ups B t d , d u shrink a subspace. Else, for some d ď σ , D T i P M at p d , F q , i “ 1 , . . . , m , det p T 1 b B 1 ` T 2 b B 2 ` . . . ` T m b B m q ‰ 0

  15. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Implications of degree bound σ Theorem [IQS15a] Over Q , if the nullcone is defined by elements of degree ď σ “ σ p n , m q ,there exists a deterministic poly( n , m , σ ) algorithm deciding if ( B 1 , B 2 , . . . , B m ) is in the null cone. If p B 1 , . . . , B m q is in the null cone all blow-ups B t d , d u shrink a subspace. Else, for some d ď σ , D T i P M at p d , F q , i “ 1 , . . . , m , det p T 1 b B 1 ` T 2 b B 2 ` . . . ` T m b B m q ‰ 0 i.e. B t d , d u contains a nonsingular matrix.

  16. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Implications of degree bound σ Theorem [IQS15a] Over Q , if the nullcone is defined by elements of degree ď σ “ σ p n , m q ,there exists a deterministic poly( n , m , σ ) algorithm deciding if ( B 1 , B 2 , . . . , B m ) is in the null cone. If p B 1 , . . . , B m q is in the null cone all blow-ups B t d , d u shrink a subspace. Else, for some d ď σ , D T i P M at p d , F q , i “ 1 , . . . , m , det p T 1 b B 1 ` T 2 b B 2 ` . . . ` T m b B m q ‰ 0 i.e. B t d , d u contains a nonsingular matrix. Gurvits promise condition is met at stage d .

  17. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Implications of degree bound σ Theorem [IQS15a] Over Q , if the nullcone is defined by elements of degree ď σ “ σ p n , m q ,there exists a deterministic poly( n , m , σ ) algorithm deciding if ( B 1 , B 2 , . . . , B m ) is in the null cone. If p B 1 , . . . , B m q is in the null cone all blow-ups B t d , d u shrink a subspace. Else, for some d ď σ , D T i P M at p d , F q , i “ 1 , . . . , m , det p T 1 b B 1 ` T 2 b B 2 ` . . . ` T m b B m q ‰ 0 i.e. B t d , d u contains a nonsingular matrix. Gurvits promise condition is met at stage d . For i “ 1 : σ run Gurvits’ algorithm on B i , i :

  18. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Implications of degree bound σ Theorem [IQS15a] Over Q , if the nullcone is defined by elements of degree ď σ “ σ p n , m q ,there exists a deterministic poly( n , m , σ ) algorithm deciding if ( B 1 , B 2 , . . . , B m ) is in the null cone. If p B 1 , . . . , B m q is in the null cone all blow-ups B t d , d u shrink a subspace. Else, for some d ď σ , D T i P M at p d , F q , i “ 1 , . . . , m , det p T 1 b B 1 ` T 2 b B 2 ` . . . ` T m b B m q ‰ 0 i.e. B t d , d u contains a nonsingular matrix. Gurvits promise condition is met at stage d . For i “ 1 : σ run Gurvits’ algorithm on B i , i : If Gurvits says Rk p B i , i q “ i ˚ n , output Rk p B q “ n ; exit.

  19. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Implications of degree bound σ Theorem [IQS15a] Over Q , if the nullcone is defined by elements of degree ď σ “ σ p n , m q ,there exists a deterministic poly( n , m , σ ) algorithm deciding if ( B 1 , B 2 , . . . , B m ) is in the null cone. If p B 1 , . . . , B m q is in the null cone all blow-ups B t d , d u shrink a subspace. Else, for some d ď σ , D T i P M at p d , F q , i “ 1 , . . . , m , det p T 1 b B 1 ` T 2 b B 2 ` . . . ` T m b B m q ‰ 0 i.e. B t d , d u contains a nonsingular matrix. Gurvits promise condition is met at stage d . For i “ 1 : σ run Gurvits’ algorithm on B i , i : If Gurvits says Rk p B i , i q “ i ˚ n , output Rk p B q “ n ; exit. Output NCrk p B q ă n .

  20. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Suggested algorithm

  21. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Suggested algorithm Can we modify the suggested algorithm suitably?

  22. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Suggested algorithm Can we modify the suggested algorithm suitably? Recall If B shrinks U , then so will its d -th blow-up.

  23. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Suggested algorithm Can we modify the suggested algorithm suitably? Recall If B shrinks U , then so will its d -th blow-up. for i “ 1 , 2 , . . . , compute (a basis of) x B t i , i u y ,

  24. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Suggested algorithm Can we modify the suggested algorithm suitably? Recall If B shrinks U , then so will its d -th blow-up. for i “ 1 , 2 , . . . , compute (a basis of) x B t i , i u y , determine if there is a nonsingular matrix in the blow-up.

  25. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Suggested algorithm Can we modify the suggested algorithm suitably? Recall If B shrinks U , then so will its d -th blow-up. for i “ 1 , 2 , . . . , compute (a basis of) x B t i , i u y , determine if there is a nonsingular matrix in the blow-up. However...finding a nonsingular matrix in the span will be difficult.

  26. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Suggested algorithm Can we modify the suggested algorithm suitably? Recall If B shrinks U , then so will its d -th blow-up. for i “ 1 , 2 , . . . , compute (a basis of) x B t i , i u y ,and a matrix M i ´ 1 . determine if there is a nonsingular matrix in the blow-up.

  27. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Suggested algorithm Can we modify the suggested algorithm suitably? Recall If B shrinks U , then so will its d -th blow-up. for i “ 1 , 2 , . . . , compute (a basis of) x B t i , i u y ,and a matrix M i ´ 1 . determine if there is a nonsingular matrix in the blow-up. Using M i ´ 1 , update and get M i , achieving some measurable progress.

  28. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Outline 1 Background and problem statement Problem statement Invariant theory 2 Using Gurvits algorithm 3 Progress via Blow-ups Regularity Algorithmic and degree bounds Degree bounds Polynomial bound - degree of generation Main lemma and blow ups using division algebras Proof of the main lemma Matrix of maximum rank

  29. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Regularity Regularity of Blow-ups Main Lemma For B ď M at p n , F q and A “ B t d , d u , assume that | F | ą 2 rd . Given a matrix A P A with rk A ą p r ´ 1 q d , there exists a deterministic algorithm that returns r A P A and an r ˆ r window W in r A s.t. W is nonsingular (of rank rd ). This algorithm uses poly p nd q operations and, over Q , the algorithm runs in polynomial time.

  30. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Regularity Regularity of Blow-ups Main Lemma For B ď M at p n , F q and A “ B t d , d u , assume that | F | ą 2 rd . Given a matrix A P A with rk A ą p r ´ 1 q d , there exists a deterministic algorithm that returns r A P A and an r ˆ r window W in r A s.t. W is nonsingular (of rank rd ). This algorithm uses poly p nd q operations and, over Q , the algorithm runs in polynomial time.

  31. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Regularity Regularity of Blow-ups Main Lemma For B ď M at p n , F q and A “ B t d , d u , assume that | F | ą 2 rd . Given a matrix A P A with rk A ą p r ´ 1 q d , there exists a deterministic algorithm that returns r A P A and an r ˆ r window W in r A s.t. W is nonsingular (of rank rd ). This algorithm uses poly p nd q operations and, over Q , the algorithm runs in polynomial time. The matrix with maximum rank in the d -th blow-up has rank a multiple of d .

  32. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Regularity Regularity of Blow-ups Main Lemma For B ď M at p n , F q and A “ B t d , d u , assume that | F | ą 2 rd . Given a matrix A P A with rk A ą p r ´ 1 q d , there exists a deterministic algorithm that returns r A P A and an r ˆ r window W in r A s.t. W is nonsingular (of rank rd ). This algorithm uses poly p nd q operations and, over Q , the algorithm runs in polynomial time. The matrix with maximum rank in the d -th blow-up has rank a multiple of d . Starting with a matrix of rank p r ´ 1 q d ` 1 in A , we construct a matrix of rank rd in A - a constructive proof.

  33. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Regularity Regularity of Blow-ups Main Lemma For B ď M at p n , F q and A “ B t d , d u , assume that | F | ą 2 rd . Given a matrix A P A with rk A ą p r ´ 1 q d , there exists a deterministic algorithm that returns r A P A and an r ˆ r window W in r A s.t. W is nonsingular (of rank rd ). This algorithm uses poly p nd q operations and, over Q , the algorithm runs in polynomial time. The matrix with maximum rank in the d -th blow-up has rank a multiple of d . Starting with a matrix of rank p r ´ 1 q d ` 1 in A , we construct a matrix of rank rd in A - a constructive proof. Central division algebras almost do our job.

  34. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Regularity Suggested algorithm

  35. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Regularity Suggested algorithm 1 Start with a matrix in the given family B of rank r .

  36. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Regularity Suggested algorithm 1 Start with a matrix in the given family B of rank r . 2 Determine if this is the matrix with largest rank in the family.

  37. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Regularity Suggested algorithm 1 Start with a matrix in the given family B of rank r . 2 Determine if this is the matrix with largest rank in the family. 3 If not, consider the r ` 1-th blow blow up A “ B r ` 1 , r ` 1 .

  38. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Regularity Suggested algorithm 1 Start with a matrix in the given family B of rank r . 2 Determine if this is the matrix with largest rank in the family. 3 If not, consider the r ` 1-th blow blow up A “ B r ` 1 , r ` 1 . 4 Starting with a rank r matrix in this blow up, find a matrix of rank at least r p r ` 1 q ` 1.

  39. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Regularity Suggested algorithm 1 Start with a matrix in the given family B of rank r . 2 Determine if this is the matrix with largest rank in the family. 3 If not, consider the r ` 1-th blow blow up A “ B r ` 1 , r ` 1 . 4 Starting with a rank r matrix in this blow up, find a matrix of rank at least r p r ` 1 q ` 1. 5 Use regularity of blow-ups to get a matrix of rank p r ` 1 q ˚ p r ` 1 q in the blow up.

  40. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Regularity Suggested algorithm 1 Start with a matrix in the given family B of rank r . 2 Determine if this is the matrix with largest rank in the family. 3 If not, consider the r ` 1-th blow blow up A “ B r ` 1 , r ` 1 . 4 Starting with a rank r matrix in this blow up, find a matrix of rank at least r p r ` 1 q ` 1. 5 Use regularity of blow-ups to get a matrix of rank p r ` 1 q ˚ p r ` 1 q in the blow up. 6 Loop back to step 2 with B “ A and r “ r ` 1.

  41. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Regularity Realizing the algorithm

  42. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Regularity Realizing the algorithm Issues to be addressed:

  43. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Regularity Realizing the algorithm Issues to be addressed: Finding if a matrix in a given family has the largest rank.

  44. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Regularity Realizing the algorithm Issues to be addressed: Finding if a matrix in a given family has the largest rank. Incrementing rank otherwise.

  45. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Regularity Realizing the algorithm Issues to be addressed: Finding if a matrix in a given family has the largest rank. Incrementing rank otherwise. Finding a matrix with rank a multiple of the blow-up factor.

  46. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Regularity Realizing the algorithm Issues to be addressed: Finding if a matrix in a given family has the largest rank. Incrementing rank otherwise. Finding a matrix with rank a multiple of the blow-up factor. Keeping the size of matrix entries polynomial.

  47. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Regularity Realizing the algorithm Issues to be addressed: Finding if a matrix in a given family has the largest rank. Incrementing rank otherwise. Finding a matrix with rank a multiple of the blow-up factor. Keeping the size of matrix entries polynomial. Blowing down matrices to keep matrix size polynomial.

  48. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Regularity Realizing the algorithm Issues to be addressed: Finding if a matrix in a given family has the largest rank. Incrementing rank otherwise. Finding a matrix with rank a multiple of the blow-up factor. Keeping the size of matrix entries polynomial. Blowing down matrices to keep matrix size polynomial. Knowing when to stop.

  49. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Regularity Realizing the algorithm Issues to be addressed: Finding if a matrix in a given family has the largest rank. Incrementing rank otherwise. Finding a matrix with rank a multiple of the blow-up factor. Keeping the size of matrix entries polynomial. Blowing down matrices to keep matrix size polynomial. Identifying the shrunk subspace, if any. Knowing when to stop.

  50. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Algorithmic and degree bounds Upper bounds

  51. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Algorithmic and degree bounds Upper bounds [Der01] Over algebraically closed fields of characteristic zero, σ “ O p n 2 4 n 2 q . The invariant ring is generated in degree β “ O p n 2 σ 2 q .

  52. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Algorithmic and degree bounds Upper bounds [Der01] Over algebraically closed fields of characteristic zero, σ “ O p n 2 4 n 2 q . The invariant ring is generated in degree β “ O p n 2 σ 2 q . [IQS15a] When F is large, a p oly p n ` 1 ! q algorithm for computing Rk p B q so σ ď n ` 1 ! . Over algebraically closed fields of char 0, β “ O p n 4 p n ` 1 ! q 2 q .

  53. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Algorithmic and degree bounds Upper bounds [Der01] Over algebraically closed fields of characteristic zero, σ “ O p n 2 4 n 2 q . The invariant ring is generated in degree β “ O p n 2 σ 2 q . [IQS15a] When F is large, a p oly p n ` 1 ! q algorithm for computing Rk p B q so σ ď n ` 1 ! . Over algebraically closed fields of char 0, β “ O p n 4 p n ` 1 ! q 2 q . [GGOW15] used the degree bound from [IQS15a] - give a polynomial time algorithm for the nullcone membership over fields of characteristic zero.

  54. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Algorithmic and degree bounds Upper bounds [Der01] Over algebraically closed fields of characteristic zero, σ “ O p n 2 4 n 2 q . The invariant ring is generated in degree β “ O p n 2 σ 2 q . [IQS15a] When F is large, a p oly p n ` 1 ! q algorithm for computing Rk p B q so σ ď n ` 1 ! . Over algebraically closed fields of char 0, β “ O p n 4 p n ` 1 ! q 2 q . [GGOW15] used the degree bound from [IQS15a] - give a polynomial time algorithm for the nullcone membership over fields of characteristic zero. [DM15] use the regularity under blow-up lemma of [IQS15a], and a convexity argument - σ ď O p n 2 q , over algebraically closed fields, β “ O p n 6 q .

  55. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Algorithmic and degree bounds Upper bounds [Der01] Over algebraically closed fields of characteristic zero, σ “ O p n 2 4 n 2 q . The invariant ring is generated in degree β “ O p n 2 σ 2 q . [IQS15a] When F is large, a p oly p n ` 1 ! q algorithm for computing Rk p B q so σ ď n ` 1 ! . Over algebraically closed fields of char 0, β “ O p n 4 p n ` 1 ! q 2 q . [GGOW15] used the degree bound from [IQS15a] - give a polynomial time algorithm for the nullcone membership over fields of characteristic zero. [DM15] use the regularity under blow-up lemma of [IQS15a], and a convexity argument - σ ď O p n 2 q , over algebraically closed fields, β “ O p n 6 q . [IQS15b] Show σ ď O p n 2 q over all large fields. Two proofs - a constructive version of [DM15] and a simple proof based on regularity under blow-up. Get the above results.

  56. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Polynomial bound - degree of generation Blow-up upper bound of n ` 1

  57. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Polynomial bound - degree of generation Blow-up upper bound of n ` 1 Generation of the invariant ring in poly( n )-degree [DM15]. If there is no nonsingular matrix in B n ` 1 , n ` 1 , then there is no nonsingular matrix in B d , d , for all d ě n ` 1. Over infinite fields the null cone is cut by invariants of degree O p n 2 q . Over Q the ring of invariants is generated in degree O p n 6 q .

  58. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Polynomial bound - degree of generation Blow-up upper bound of n ` 1 Generation of the invariant ring in poly( n )-degree [DM15]. If there is no nonsingular matrix in B n ` 1 , n ` 1 , then there is no nonsingular matrix in B d , d , for all d ě n ` 1. Over infinite fields the null cone is cut by invariants of degree O p n 2 q . Over Q the ring of invariants is generated in degree O p n 6 q . Proof [IQS15b]

  59. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Polynomial bound - degree of generation Blow-up upper bound of n ` 1 Generation of the invariant ring in poly( n )-degree [DM15]. If there is no nonsingular matrix in B n ` 1 , n ` 1 , then there is no nonsingular matrix in B d , d , for all d ě n ` 1. Over infinite fields the null cone is cut by invariants of degree O p n 2 q . Over Q the ring of invariants is generated in degree O p n 6 q . Proof [IQS15b] Take d “ n ` 2.

  60. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Polynomial bound - degree of generation Blow-up upper bound of n ` 1 Generation of the invariant ring in poly( n )-degree [DM15]. If there is no nonsingular matrix in B n ` 1 , n ` 1 , then there is no nonsingular matrix in B d , d , for all d ě n ` 1. Over infinite fields the null cone is cut by invariants of degree O p n 2 q . Over Q the ring of invariants is generated in degree O p n 6 q . Proof [IQS15b] Take d “ n ` 2. So the largest ranked matrix in a n ` 1 ˆ n ` 1 window is p n ` 1 q ˚ p n ´ 1 q “ n 2 ´ 1.

  61. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Polynomial bound - degree of generation Blow-up upper bound of n ` 1 Generation of the invariant ring in poly( n )-degree [DM15]. If there is no nonsingular matrix in B n ` 1 , n ` 1 , then there is no nonsingular matrix in B d , d , for all d ě n ` 1. Over infinite fields the null cone is cut by invariants of degree O p n 2 q . Over Q the ring of invariants is generated in degree O p n 6 q . Proof [IQS15b] Take d “ n ` 2. So the largest ranked matrix in a n ` 1 ˆ n ` 1 window is p n ` 1 q ˚ p n ´ 1 q “ n 2 ´ 1. But we add to such a matrix at most 2 n linearly independent rows and columns.

  62. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Polynomial bound - degree of generation Blow-up upper bound of n ` 1 Generation of the invariant ring in poly( n )-degree [DM15]. If there is no nonsingular matrix in B n ` 1 , n ` 1 , then there is no nonsingular matrix in B d , d , for all d ě n ` 1. Over infinite fields the null cone is cut by invariants of degree O p n 2 q . Over Q the ring of invariants is generated in degree O p n 6 q . Proof [IQS15b] Take d “ n ` 2. So the largest ranked matrix in a n ` 1 ˆ n ` 1 window is p n ` 1 q ˚ p n ´ 1 q “ n 2 ´ 1. But we add to such a matrix at most 2 n linearly independent rows and columns. So rank is upper bounded by n 2 ´ 1 ` 2 n , .

  63. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Polynomial bound - degree of generation Blow-up upper bound of n ` 1 Generation of the invariant ring in poly( n )-degree [DM15]. If there is no nonsingular matrix in B n ` 1 , n ` 1 , then there is no nonsingular matrix in B d , d , for all d ě n ` 1. Over infinite fields the null cone is cut by invariants of degree O p n 2 q . Over Q the ring of invariants is generated in degree O p n 6 q . Proof [IQS15b] Take d “ n ` 2. So the largest ranked matrix in a n ` 1 ˆ n ` 1 window is p n ` 1 q ˚ p n ´ 1 q “ n 2 ´ 1. But we add to such a matrix at most 2 n linearly independent rows and columns. So rank is upper bounded by n 2 ´ 1 ` 2 n , cannot be p n ` 2 q ˚ n .

  64. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Polynomial bound - degree of generation Blow-up upper bound of n ` 1 Generation of the invariant ring in poly( n )-degree [DM15]. If there is no nonsingular matrix in B n ` 1 , n ` 1 , then there is no nonsingular matrix in B d , d , for all d ě n ` 1. Over infinite fields the null cone is cut by invariants of degree O p n 2 q . Over Q the ring of invariants is generated in degree O p n 6 q . Proof [IQS15b] Take d “ n ` 2. So the largest ranked matrix in a n ` 1 ˆ n ` 1 window is p n ` 1 q ˚ p n ´ 1 q “ n 2 ´ 1. But we add to such a matrix at most 2 n linearly independent rows and columns. So rank is upper bounded by n 2 ´ 1 ` 2 n , cannot be p n ` 2 q ˚ n . Regularity says rank is at most p n ` 2 q ˚ p n ´ 1 q “ n 2 ` n ´ 2. QED

  65. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Main lemma and blow ups using division algebras Blowing-up using a division algebra.

  66. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Main lemma and blow ups using division algebras Blowing-up using a division algebra. Claim Let F 1 be an extension field of F , and Let D be a central division algebra over F 1 of dimension d 2 over F 1 , and let K be a maximal field in D with extension degree d over F 1 . Let ρ : D Ñ M at p d , K q be a representation of D over K . Then every matrix in M at p n , F q b F ρ p D q has rank divisible by d over K .

  67. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Main lemma and blow ups using division algebras Blowing-up using a division algebra. Claim Let F 1 be an extension field of F , and Let D be a central division algebra over F 1 of dimension d 2 over F 1 , and let K be a maximal field in D with extension degree d over F 1 . Let ρ : D Ñ M at p d , K q be a representation of D over K . Then every matrix in M at p n , F q b F ρ p D q has rank divisible by d over K . D b K – M at p K q . Explicit matrices describing the F 1 -algebra D – D b 1 can be written down easily.

  68. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Main lemma and blow ups using division algebras Blowing-up using a division algebra. Claim Let F 1 be an extension field of F , and Let D be a central division algebra over F 1 of dimension d 2 over F 1 , and let K be a maximal field in D with extension degree d over F 1 . Let ρ : D Ñ M at p d , K q be a representation of D over K . Then every matrix in M at p n , F q b F ρ p D q has rank divisible by d over K . D b K – M at p K q . Explicit matrices describing the F 1 -algebra D – D b 1 can be written down easily. Regard K dn – F 1 d 2 n as a module over M at p n , F q b F ρ p D q .

  69. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Main lemma and blow ups using division algebras Blowing-up using a division algebra. Claim Let F 1 be an extension field of F , and Let D be a central division algebra over F 1 of dimension d 2 over F 1 , and let K be a maximal field in D with extension degree d over F 1 . Let ρ : D Ñ M at p d , K q be a representation of D over K . Then every matrix in M at p n , F q b F ρ p D q has rank divisible by d over K . D b K – M at p K q . Explicit matrices describing the F 1 -algebra D – D b 1 can be written down easily. Regard K dn – F 1 d 2 n as a module over M at p n , F q b F ρ p D q . Since D b D op – M at p d , F 1 q Ă M at p K q , the centralizer of the action of M at p n , F q b F ρ p D q is i d b D op – D op .

  70. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Main lemma and blow ups using division algebras Blowing-up using a division algebra. Claim Let F 1 be an extension field of F , and Let D be a central division algebra over F 1 of dimension d 2 over F 1 , and let K be a maximal field in D with extension degree d over F 1 . Let ρ : D Ñ M at p d , K q be a representation of D over K . Then every matrix in M at p n , F q b F ρ p D q has rank divisible by d over K . D b K – M at p K q . Explicit matrices describing the F 1 -algebra D – D b 1 can be written down easily. Regard K dn – F 1 d 2 n as a module over M at p n , F q b F ρ p D q . Since D b D op – M at p d , F 1 q Ă M at p K q , the centralizer of the action of M at p n , F q b F ρ p D q is i d b D op – D op . For all A in M at p n , F q b F ρ p D q , A F 1 d 2 n is a D op -submodule, and so its dimension over F 1 is divisible by d 2 , so dimension over K is divisible by d . But this is the rank of A 1 .

  71. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Proof of the main lemma Recap Main Lemma For B ď M at p n , F q and A “ B t d , d u , assume that | F | ą 2 rd . Given a matrix A P A with rk A ą p r ´ 1 q d , there exists a deterministic algorithm that returns r A P A and an r ˆ r window W in r A s.t. W is nonsingular (of rank rd ). This algorithm uses poly p nd q operations and, over Q , the algorithm runs in polynomial time.

  72. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Proof of the main lemma Proof

  73. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Proof of the main lemma Proof Assuming we have a division algebra and a representation of it.

  74. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Proof of the main lemma Proof Assuming we have a division algebra and a representation of it. Induction on r : Base case: r “ 1 - there is at least one nonzero matrix B in B ; p i , j q -th entry is nonzero then we have a d ˆ d block in B b I which is non zero, of rank d .

  75. Background and problem statement Using Gurvits algorithm Progress via Blow-ups Proof of the main lemma Proof Assuming we have a division algebra and a representation of it. Induction on r : Base case: r “ 1 - there is at least one nonzero matrix B in B ; p i , j q -th entry is nonzero then we have a d ˆ d block in B b I which is non zero, of rank d . By induction, the principal p r ´ 1 q window of A 1 P A “ B t d , d u has non-zero determinant.

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