Null cone membership for the left right action on tuples of matrices - - PowerPoint PPT Presentation

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

Null cone membership for the left right action on tuples of matrices

Gabor Ivanyos1, Jimmy Qiao2, K V Subrahmanyam3

1Institute for Computer Science and Control, Hungarian Academy of Sciences,

Budapest

2Center for Quantum Computation and Intelligent Systems, Univ of Technology,

Sydney

3Chennai Mathematical Institute, Chennai

Tel Aviv University, Feb 09, 2016

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 Division algebras

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

Background and problem statement Using Gurvits algorithm Progress via Blow-ups Problem statement

Matpn, Fq - n ˆ n matrices with entries in F.

Background and problem statement Using Gurvits algorithm Progress via Blow-ups Problem statement

Matpn, Fq - n ˆ n matrices with entries in F. B1, B2, . . . , Bm P Matpn, Fq.

Background and problem statement Using Gurvits algorithm Progress via Blow-ups Problem statement

Matpn, Fq - n ˆ n matrices with entries in F. B1, B2, . . . , Bm P Matpn, Fq. B - F-linear span of the matrices x B1, B2, . . . , Bmy.

Background and problem statement Using Gurvits algorithm Progress via Blow-ups Problem statement

Matpn, Fq - n ˆ n matrices with entries in F. B1, B2, . . . , Bm P Matpn, Fq. B - F-linear span of the matrices x B1, B2, . . . , Bmy. Shrunk subspaces A subspace U ď Fn is c-shrunk by B if there is a subspace W Ď Fn such that dim W ď dimU ´ c, and for all matrices B in B, xBUy Ď W.

Background and problem statement Using Gurvits algorithm Progress via Blow-ups Problem statement

Matpn, Fq - n ˆ n matrices with entries in F. B1, B2, . . . , Bm P Matpn, Fq. B - F-linear span of the matrices x B1, B2, . . . , Bmy. Shrunk subspaces A subspace U ď Fn is c-shrunk by B if there is a subspace W Ď Fn such that dim W ď dimU ´ c, and for all matrices B in B, xBUy Ď W. Non commutative rank n ´ maxpc P t0, 1, . . . , nu | Dsubspace c-shrunk by Bq [FR04].

Background and problem statement Using Gurvits algorithm Progress via Blow-ups Problem statement

Matpn, Fq - n ˆ n matrices with entries in F. B1, B2, . . . , Bm P Matpn, Fq. B - F-linear span of the matrices x B1, B2, . . . , Bmy. Shrunk subspaces A subspace U ď Fn is c-shrunk by B if there is a subspace W Ď Fn such that dim W ď dimU ´ c, and for all matrices B in B, xBUy Ď W. Non commutative rank n ´ maxpc P t0, 1, . . . , nu | Dsubspace c-shrunk by Bq [FR04]. Problem NCrk: What is the noncommutative rank?

Background and problem statement Using Gurvits algorithm Progress via Blow-ups Problem statement

Related problem

Commutative rank The maximum of the rank of matrices in B?

Background and problem statement Using Gurvits algorithm Progress via Blow-ups Problem statement

Related problem

Commutative rank The maximum of the rank of matrices in B? Problem Rk: What is the commutative rank? [Edm67]

Background and problem statement Using Gurvits algorithm Progress via Blow-ups Problem statement

Related problem

Commutative rank The maximum of the rank of matrices in B? Problem Rk: What is the commutative rank? [Edm67] Rk ď NCrk.

Background and problem statement Using Gurvits algorithm Progress via Blow-ups Problem statement

Related problem

Commutative rank The maximum of the rank of matrices in B? Problem Rk: What is the commutative rank? [Edm67] Rk ď NCrk. For the family of 3 ˆ 3 skew symmetric matrices, 2=Rk < NCrk=3.

Background and problem statement Using Gurvits algorithm Progress via Blow-ups Problem statement

Related problem

Commutative rank The maximum of the rank of matrices in B? Problem Rk: What is the commutative rank? [Edm67] Rk ď NCrk. For the family of 3 ˆ 3 skew symmetric matrices, 2=Rk < NCrk=3. Theorem - Gurvits Over Q, given a matrix space xBy there is a deterministic polynomial time algorithm which will output Rk=n, or NCrk ă n, and its output is guaranteed to be correct when either NCrkpBq ă n or RkpBq “ n.

Background and problem statement Using Gurvits algorithm Progress via Blow-ups Problem statement

Related problem

Commutative rank The maximum of the rank of matrices in B? Problem Rk: What is the commutative rank? [Edm67] Rk ď NCrk. For the family of 3 ˆ 3 skew symmetric matrices, 2=Rk < NCrk=3. Theorem - Gurvits Over Q, given a matrix space xBy there is a deterministic polynomial time algorithm which will output Rk=n, or NCrk ă n, and its output is guaranteed to be correct when either NCrkpBq ă n or RkpBq “ n. The algorithm may give a wrong answer in the case when n “ NCrk ą Rk.

Background and problem statement Using Gurvits algorithm Progress via Blow-ups Invariant theory

Left right action

X = { X1, X2, . . . , Xm}, Xk, an n ˆ n matrix with variable entries xk

ij .

Background and problem statement Using Gurvits algorithm Progress via Blow-ups Invariant theory

Left right action

X = { X1, X2, . . . , Xm}, Xk, an n ˆ n matrix with variable entries xk

ij .

SLn ˆ SLn ñ X,

Background and problem statement Using Gurvits algorithm Progress via Blow-ups Invariant theory

Left right action

X = { X1, X2, . . . , Xm}, Xk, an n ˆ n matrix with variable entries xk

ij .

SLn ˆ SLn ñ X, pA, Bq ¨ { X1, X2, . . . , Xm} = { AX1Bt, AX2Bt, . . . , AXmBt}.

Background and problem statement Using Gurvits algorithm Progress via Blow-ups Invariant theory

Left right action

X = { X1, X2, . . . , Xm}, Xk, an n ˆ n matrix with variable entries xk

ij .

SLn ˆ SLn ñ X, pA, Bq ¨ { X1, X2, . . . , Xm} = { AX1Bt, AX2Bt, . . . , AXmBt}. Classical invariant theory questions What are the polynomial functions invariant under the action? - The ring of invariants is known to be finitely generated - bound on the degree in which this is generated?

Background and problem statement Using Gurvits algorithm Progress via Blow-ups Invariant theory

Left right action

X = { X1, X2, . . . , Xm}, Xk, an n ˆ n matrix with variable entries xk

ij .

SLn ˆ SLn ñ X, pA, Bq ¨ { X1, X2, . . . , Xm} = { AX1Bt, AX2Bt, . . . , AXmBt}. Classical invariant theory questions What are the polynomial functions invariant under the action? - well understood characteristic zero fields,[Sch91, DZ01, ANS07], infinite fields [DZ01]. The ring of invariants is known to be finitely generated - bound on the degree in which this is generated?

Background and problem statement Using Gurvits algorithm Progress via Blow-ups Invariant theory

Left right action

X = { X1, X2, . . . , Xm}, Xk, an n ˆ n matrix with variable entries xk

ij .

SLn ˆ SLn ñ X, pA, Bq ¨ { X1, X2, . . . , Xm} = { AX1Bt, AX2Bt, . . . , AXmBt}. Classical invariant theory questions What are the polynomial functions invariant under the action? - well understood characteristic zero fields,[Sch91, DZ01, ANS07], infinite fields [DZ01]. The ring of invariants is known to be finitely generated - bound on the degree in which this is generated? characteristic zero fields, exppn2q, [Der01].

Background and problem statement Using Gurvits algorithm Progress via Blow-ups Invariant theory

Membership in the null cone

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

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 fpB1, B2, . . . , Bmq “ 0 for all invariant polynomial functions f.

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 fpB1, B2, . . . , Bmq “ 0 for all invariant polynomial functions f. Over infinite fields - an alternate characterization -

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 fpB1, B2, . . . , Bmq “ 0 for all invariant polynomial functions f. Over infinite fields - an alternate characterization - (B1, B2, . . . , Bm) such that B has a c-shrunk subspace for c ą 0 [BD06, DZ01, ANS07].

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 fpB1, B2, . . . , Bmq “ 0 for all invariant polynomial functions f. Over infinite fields - an alternate characterization - (B1, B2, . . . , Bm) such that B has a c-shrunk subspace for c ą 0 [BD06, DZ01, ANS07]. A description of the invariants: Let T1, T2, . . . , Tm be matrices in Matpd, Fq. Then detpT1 b X1 ` T2 b X2 ` . . . ` Tm b Xmq is an invariant of degree nd.

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 fpB1, B2, . . . , Bmq “ 0 for all invariant polynomial functions f. Over infinite fields - an alternate characterization - (B1, B2, . . . , Bm) such that B has a c-shrunk subspace for c ą 0 [BD06, DZ01, ANS07]. A description of the invariants: Let T1, T2, . . . , Tm be matrices in Matpd, Fq. Then detpT1 b X1 ` T2 b X2 ` . . . ` Tm b Xmq is an invariant of degree nd. Over infinite fields, all invariants are obtained this way.

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

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

Suggested algorithm

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

Suggested algorithm

Observation If B1 shrinks a subspace U P Fn, and T1 P Matpd, Fq then T1 b B1 shrinks the subspace U b Fd.

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

Suggested algorithm

Observation If B1 shrinks a subspace U P Fn, and T1 P Matpd, Fq then T1 b B1 shrinks the subspace U b Fd. If B shrinks U, then so will its d-th blow-up

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

Suggested algorithm

Observation If B1 shrinks a subspace U P Fn, and T1 P Matpd, Fq then T1 b B1 shrinks the subspace U b Fd. If B shrinks U, then so will its d-th blow-up Btd,du:=x T1 b B1, T2 b B2, . . . , Tm b Bmy, Ti P Matpd, Fq.

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

Suggested algorithm

Observation If B1 shrinks a subspace U P Fn, and T1 P Matpd, Fq then T1 b B1 shrinks the subspace U b Fd. If B shrinks U, then so will its d-th blow-up Btd,du:=x T1 b B1, T2 b B2, . . . , Tm b Bmy, Ti P Matpd, Fq. for i “ 1, 2, . . . , compute (a basis of) Bti,iu.

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

Suggested algorithm

Observation If B1 shrinks a subspace U P Fn, and T1 P Matpd, Fq then T1 b B1 shrinks the subspace U b Fd. If B shrinks U, then so will its d-th blow-up Btd,du:=x T1 b B1, T2 b B2, . . . , Tm b Bmy, Ti P Matpd, Fq. for i “ 1, 2, . . . , compute (a basis of) Bti,iu. determine if there is a nonsingular matrix in the blow-up.

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

Suggested algorithm

Observation If B1 shrinks a subspace U P Fn, and T1 P Matpd, Fq then T1 b B1 shrinks the subspace U b Fd. If B shrinks U, then so will its d-th blow-up Btd,du:=x T1 b B1, T2 b B2, . . . , Tm b Bmy, Ti P Matpd, Fq. for i “ 1, 2, . . . , compute (a basis of) Bti,iu. determine if there is a nonsingular matrix in the blow-up. Question How long do we go on?

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 “ σpn, mq,there exists a deterministic poly(n, m, σ) algorithm deciding if (B1, B2, . . . , Bm) is in the null cone.

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 “ σpn, mq,there exists a deterministic poly(n, m, σ) algorithm deciding if (B1, B2, . . . , Bm) is in the null cone. If pB1, . . . , Bmq is in the null cone all blow-ups Btd,du shrink a subspace.

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 “ σpn, mq,there exists a deterministic poly(n, m, σ) algorithm deciding if (B1, B2, . . . , Bm) is in the null cone. If pB1, . . . , Bmq is in the null cone all blow-ups Btd,du shrink a subspace. Else, for some d ď σ, DTi P Matpd, Fq, i “ 1, . . . , m, detpT1 b B1 ` T2 b B2 ` . . . ` Tm b Bmq ‰ 0

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 “ σpn, mq,there exists a deterministic poly(n, m, σ) algorithm deciding if (B1, B2, . . . , Bm) is in the null cone. If pB1, . . . , Bmq is in the null cone all blow-ups Btd,du shrink a subspace. Else, for some d ď σ, DTi P Matpd, Fq, i “ 1, . . . , m, detpT1 b B1 ` T2 b B2 ` . . . ` Tm b Bmq ‰ 0 i.e. Btd,du contains a nonsingular matrix.

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 “ σpn, mq,there exists a deterministic poly(n, m, σ) algorithm deciding if (B1, B2, . . . , Bm) is in the null cone. If pB1, . . . , Bmq is in the null cone all blow-ups Btd,du shrink a subspace. Else, for some d ď σ, DTi P Matpd, Fq, i “ 1, . . . , m, detpT1 b B1 ` T2 b B2 ` . . . ` Tm b Bmq ‰ 0 i.e. Btd,du contains a nonsingular matrix. Gurvits promise condition is met at stage d.

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 “ σpn, mq,there exists a deterministic poly(n, m, σ) algorithm deciding if (B1, B2, . . . , Bm) is in the null cone. If pB1, . . . , Bmq is in the null cone all blow-ups Btd,du shrink a subspace. Else, for some d ď σ, DTi P Matpd, Fq, i “ 1, . . . , m, detpT1 b B1 ` T2 b B2 ` . . . ` Tm b Bmq ‰ 0 i.e. Btd,du contains a nonsingular matrix. Gurvits promise condition is met at stage d. For i “ 1 : σ run Gurvits’ algorithm on Bi,i :

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 “ σpn, mq,there exists a deterministic poly(n, m, σ) algorithm deciding if (B1, B2, . . . , Bm) is in the null cone. If pB1, . . . , Bmq is in the null cone all blow-ups Btd,du shrink a subspace. Else, for some d ď σ, DTi P Matpd, Fq, i “ 1, . . . , m, detpT1 b B1 ` T2 b B2 ` . . . ` Tm b Bmq ‰ 0 i.e. Btd,du contains a nonsingular matrix. Gurvits promise condition is met at stage d. For i “ 1 : σ run Gurvits’ algorithm on Bi,i : If Gurvits says RkpBi,iq “ i ˚ n, output RkpBq “ n; exit.

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 “ σpn, mq,there exists a deterministic poly(n, m, σ) algorithm deciding if (B1, B2, . . . , Bm) is in the null cone. If pB1, . . . , Bmq is in the null cone all blow-ups Btd,du shrink a subspace. Else, for some d ď σ, DTi P Matpd, Fq, i “ 1, . . . , m, detpT1 b B1 ` T2 b B2 ` . . . ` Tm b Bmq ‰ 0 i.e. Btd,du contains a nonsingular matrix. Gurvits promise condition is met at stage d. For i “ 1 : σ run Gurvits’ algorithm on Bi,i : If Gurvits says RkpBi,iq “ i ˚ n, output RkpBq “ n; exit. Output NCrkpBq ă n.

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

Suggested algorithm

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

Suggested algorithm

Can we modify the suggested algorithm suitably?

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.

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 Bti,iuy,

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 Bti,iuy, determine if there is a nonsingular matrix in the blow-up.

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 Bti,iuy, determine if there is a nonsingular matrix in the blow-up. However...finding a nonsingular matrix in the span will be difficult.

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 Bti,iuy,and a matrix Mi´1. determine if there is a nonsingular matrix in the blow-up.

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 Bti,iuy,and a matrix Mi´1. determine if there is a nonsingular matrix in the blow-up. Using Mi´1, update and get Mi, achieving some measurable progress.

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

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

Regularity of Blow-ups

Main Lemma For B ď Matpn, Fq and A “ Btd,du, assume that |F| ą 2rd. Given a matrix A P A with rkA ą pr ´ 1qd, 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 polypndq operations and, over Q, the algorithm runs in polynomial time.

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

Regularity of Blow-ups

Main Lemma For B ď Matpn, Fq and A “ Btd,du, assume that |F| ą 2rd. Given a matrix A P A with rkA ą pr ´ 1qd, 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 polypndq operations and, over Q, the algorithm runs in polynomial time.

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

Regularity of Blow-ups

Main Lemma For B ď Matpn, Fq and A “ Btd,du, assume that |F| ą 2rd. Given a matrix A P A with rkA ą pr ´ 1qd, 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 polypndq 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.

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

Regularity of Blow-ups

Main Lemma For B ď Matpn, Fq and A “ Btd,du, assume that |F| ą 2rd. Given a matrix A P A with rkA ą pr ´ 1qd, 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 polypndq 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 pr ´ 1qd ` 1 in A, we construct a matrix of rank rd in A - a constructive proof.

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

Regularity of Blow-ups

Main Lemma For B ď Matpn, Fq and A “ Btd,du, assume that |F| ą 2rd. Given a matrix A P A with rkA ą pr ´ 1qd, 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 polypndq 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 pr ´ 1qd ` 1 in A, we construct a matrix of rank rd in A - a constructive proof. Central division algebras almost do our job.

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

Suggested algorithm

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.

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.

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 “ Br`1,r`1.

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 “ Br`1,r`1. 4 Starting with a rank r matrix in this blow up, find a matrix of rank at least rpr ` 1q ` 1.

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 “ Br`1,r`1. 4 Starting with a rank r matrix in this blow up, find a matrix of rank at least rpr ` 1q ` 1. 5 Use regularity of blow-ups to get a matrix of rank pr ` 1q ˚ pr ` 1q in the blow up.

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 “ Br`1,r`1. 4 Starting with a rank r matrix in this blow up, find a matrix of rank at least rpr ` 1q ` 1. 5 Use regularity of blow-ups to get a matrix of rank pr ` 1q ˚ pr ` 1q in the blow up. 6 Loop back to step 2 with B “ A and r “ r ` 1.

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

Realizing the algorithm

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

Realizing the algorithm

Issues to be addressed:

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.

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.

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.

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.

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.

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.

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.

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

Upper bounds

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 “ Opn24n2q. The invariant ring is generated in degree β “ Opn2σ2q.

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 “ Opn24n2q. The invariant ring is generated in degree β “ Opn2σ2q. [IQS15a] When F is large, a polypn ` 1!q algorithm for computing RkpBq so σ ď n ` 1!. Over algebraically closed fields of char 0, β “ Opn4pn ` 1!q2q.

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 “ Opn24n2q. The invariant ring is generated in degree β “ Opn2σ2q. [IQS15a] When F is large, a polypn ` 1!q algorithm for computing RkpBq so σ ď n ` 1!. Over algebraically closed fields of char 0, β “ Opn4pn ` 1!q2q. [GGOW15] used the degree bound from [IQS15a] - give a polynomial time algorithm for the nullcone membership

• ver fields of characteristic zero.

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 “ Opn24n2q. The invariant ring is generated in degree β “ Opn2σ2q. [IQS15a] When F is large, a polypn ` 1!q algorithm for computing RkpBq so σ ď n ` 1!. Over algebraically closed fields of char 0, β “ Opn4pn ` 1!q2q. [GGOW15] used the degree bound from [IQS15a] - give a polynomial time algorithm for the nullcone membership

• ver fields of characteristic zero.

[DM15] use the regularity under blow-up lemma of [IQS15a], and a convexity argument - σ ď Opn2q, over algebraically closed fields, β “ Opn6q.

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 “ Opn24n2q. The invariant ring is generated in degree β “ Opn2σ2q. [IQS15a] When F is large, a polypn ` 1!q algorithm for computing RkpBq so σ ď n ` 1!. Over algebraically closed fields of char 0, β “ Opn4pn ` 1!q2q. [GGOW15] used the degree bound from [IQS15a] - give a polynomial time algorithm for the nullcone membership

• ver fields of characteristic zero.

[DM15] use the regularity under blow-up lemma of [IQS15a], and a convexity argument - σ ď Opn2q, over algebraically closed fields, β “ Opn6q. [IQS15b] Show σ ď Opn2q 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.

Background and problem statement Using Gurvits algorithm Progress via Blow-ups Polynomial bound - degree of generation

Blow-up upper bound of n ` 1

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 Bn`1,n`1, then there is no nonsingular matrix in Bd,d, for all d ě n ` 1. Over infinite fields the null cone is cut by invariants of degree Opn2q. Over Q the ring of invariants is generated in degree Opn6q.

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 Bn`1,n`1, then there is no nonsingular matrix in Bd,d, for all d ě n ` 1. Over infinite fields the null cone is cut by invariants of degree Opn2q. Over Q the ring of invariants is generated in degree Opn6q. Proof [IQS15b]

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 Bn`1,n`1, then there is no nonsingular matrix in Bd,d, for all d ě n ` 1. Over infinite fields the null cone is cut by invariants of degree Opn2q. Over Q the ring of invariants is generated in degree Opn6q. Proof [IQS15b] Take d “ n ` 2.

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 Bn`1,n`1, then there is no nonsingular matrix in Bd,d, for all d ě n ` 1. Over infinite fields the null cone is cut by invariants of degree Opn2q. Over Q the ring of invariants is generated in degree Opn6q. Proof [IQS15b] Take d “ n ` 2. So the largest ranked matrix in a n ` 1 ˆ n ` 1 window is pn ` 1q ˚ pn ´ 1q “ n2 ´ 1.

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 Bn`1,n`1, then there is no nonsingular matrix in Bd,d, for all d ě n ` 1. Over infinite fields the null cone is cut by invariants of degree Opn2q. Over Q the ring of invariants is generated in degree Opn6q. Proof [IQS15b] Take d “ n ` 2. So the largest ranked matrix in a n ` 1 ˆ n ` 1 window is pn ` 1q ˚ pn ´ 1q “ n2 ´ 1. But we add to such a matrix at most 2n linearly independent rows and columns.

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 Bn`1,n`1, then there is no nonsingular matrix in Bd,d, for all d ě n ` 1. Over infinite fields the null cone is cut by invariants of degree Opn2q. Over Q the ring of invariants is generated in degree Opn6q. Proof [IQS15b] Take d “ n ` 2. So the largest ranked matrix in a n ` 1 ˆ n ` 1 window is pn ` 1q ˚ pn ´ 1q “ n2 ´ 1. But we add to such a matrix at most 2n linearly independent rows and columns. So rank is upper bounded by n2 ´ 1 ` 2n, .

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 Bn`1,n`1, then there is no nonsingular matrix in Bd,d, for all d ě n ` 1. Over infinite fields the null cone is cut by invariants of degree Opn2q. Over Q the ring of invariants is generated in degree Opn6q. Proof [IQS15b] Take d “ n ` 2. So the largest ranked matrix in a n ` 1 ˆ n ` 1 window is pn ` 1q ˚ pn ´ 1q “ n2 ´ 1. But we add to such a matrix at most 2n linearly independent rows and columns. So rank is upper bounded by n2 ´ 1 ` 2n, cannot be pn ` 2q ˚ n.

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 Bn`1,n`1, then there is no nonsingular matrix in Bd,d, for all d ě n ` 1. Over infinite fields the null cone is cut by invariants of degree Opn2q. Over Q the ring of invariants is generated in degree Opn6q. Proof [IQS15b] Take d “ n ` 2. So the largest ranked matrix in a n ` 1 ˆ n ` 1 window is pn ` 1q ˚ pn ´ 1q “ n2 ´ 1. But we add to such a matrix at most 2n linearly independent rows and columns. So rank is upper bounded by n2 ´ 1 ` 2n, cannot be pn ` 2q ˚ n. Regularity says rank is at most pn ` 2q ˚ pn ´ 1q “ n2 ` n ´ 2. QED

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.

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 F1 be an extension field of F, and Let D be a central division algebra over F1 of dimension d2 over F1, and let K be a maximal field in D with extension degree d over F1. Let ρ : D Ñ Matpd, Kq be a representation of D over K. Then every matrix in Matpn, Fq bF ρpDq has rank divisible by d over K.

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 F1 be an extension field of F, and Let D be a central division algebra over F1 of dimension d2 over F1, and let K be a maximal field in D with extension degree d over F1. Let ρ : D Ñ Matpd, Kq be a representation of D over K. Then every matrix in Matpn, Fq bF ρpDq has rank divisible by d over K. D b K – MatpKq. Explicit matrices describing the F1-algebra D – D b 1 can be written down easily.

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 F1 be an extension field of F, and Let D be a central division algebra over F1 of dimension d2 over F1, and let K be a maximal field in D with extension degree d over F1. Let ρ : D Ñ Matpd, Kq be a representation of D over K. Then every matrix in Matpn, Fq bF ρpDq has rank divisible by d over K. D b K – MatpKq. Explicit matrices describing the F1-algebra D – D b 1 can be written down easily. Regard Kdn – F1d2n as a module over Matpn, Fq bF ρpDq.

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 F1 be an extension field of F, and Let D be a central division algebra over F1 of dimension d2 over F1, and let K be a maximal field in D with extension degree d over F1. Let ρ : D Ñ Matpd, Kq be a representation of D over K. Then every matrix in Matpn, Fq bF ρpDq has rank divisible by d over K. D b K – MatpKq. Explicit matrices describing the F1-algebra D – D b 1 can be written down easily. Regard Kdn – F1d2n as a module over Matpn, Fq bF ρpDq. Since D b Dop – Matpd, F1q Ă MatpKq, the centralizer of the action of Matpn, Fq bF ρpDq is id b Dop – Dop.

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 F1 be an extension field of F, and Let D be a central division algebra over F1 of dimension d2 over F1, and let K be a maximal field in D with extension degree d over F1. Let ρ : D Ñ Matpd, Kq be a representation of D over K. Then every matrix in Matpn, Fq bF ρpDq has rank divisible by d over K. D b K – MatpKq. Explicit matrices describing the F1-algebra D – D b 1 can be written down easily. Regard Kdn – F1d2n as a module over Matpn, Fq bF ρpDq. Since D b Dop – Matpd, F1q Ă MatpKq, the centralizer of the action of Matpn, Fq bF ρpDq is id b Dop – Dop. For all A in Matpn, Fq bF ρpDq, AF1d2n is a Dop-submodule, and so its dimension over F1 is divisible by d2, so dimension over K is divisible by d. But this is the rank of A1.

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

Recap

Main Lemma For B ď Matpn, Fq and A “ Btd,du, assume that |F| ą 2rd. Given a matrix A P A with rkA ą pr ´ 1qd, 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 polypndq operations and, over Q, the algorithm runs in polynomial time.

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

Proof

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

• f it.

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

• f it.

Induction on r: Base case: r “ 1 - there is at least one nonzero matrix B in B; pi, jq-th entry is nonzero then we have a d ˆ d block in B b I which is non zero, of rank d.

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

• f it.

Induction on r: Base case: r “ 1 - there is at least one nonzero matrix B in B; pi, jq-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 pr ´ 1q window of A1 P A “ Btd,du has non-zero determinant.

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

• f it.

Induction on r: Base case: r “ 1 - there is at least one nonzero matrix B in B; pi, jq-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 pr ´ 1q window of A1 P A “ Btd,du has non-zero determinant. Dλ, µ, with the principal r ´ 1 window of λ ˚ A ` µA1 having non-zero determinant and the principal r-window having rank at least pr ´ 1qd ` 1.

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

• f it.

Induction on r: Base case: r “ 1 - there is at least one nonzero matrix B in B; pi, jq-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 pr ´ 1q window of A1 P A “ Btd,du has non-zero determinant. Dλ, µ, with the principal r ´ 1 window of λ ˚ A ` µA1 having non-zero determinant and the principal r-window having rank at least pr ´ 1qd ` 1. Wlog we have matrix of rank at least pn ´ 1qd ` 1 with the principal n ´ 1 window having a nonsingular matrix.

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

Proof of main lemma .....

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

Proof of main lemma .....

Let ρ : D Ñ Matpd, Kq, be a representation of D.

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

Proof of main lemma .....

Let ρ : D Ñ Matpd, Kq, be a representation of D. A1 :“ A b Matpd, Kq. Then A1 “ B b Matpd, Kq is a K-linear subspace of Matpn, Kq b Matpd, Kq.

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

Proof of main lemma .....

Let ρ : D Ñ Matpd, Kq, be a representation of D. A1 :“ A b Matpd, Kq. Then A1 “ B b Matpd, Kq is a K-linear subspace of Matpn, Kq b Matpd, Kq. B bF ρpDq is an F1 linear space. Its K linear span is A1.

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

Proof of main lemma .....

Let ρ : D Ñ Matpd, Kq, be a representation of D. A1 :“ A b Matpd, Kq. Then A1 “ B b Matpd, Kq is a K-linear subspace of Matpn, Kq b Matpd, Kq. B bF ρpDq is an F1 linear space. Its K linear span is A1. Starting with the matrix A, get a matrix ˜ A in B bF ρpDq of the same rank, so rank is at least pn ´ 1qd ` 1.

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

Proof of main lemma .....

Let ρ : D Ñ Matpd, Kq, be a representation of D. A1 :“ A b Matpd, Kq. Then A1 “ B b Matpd, Kq is a K-linear subspace of Matpn, Kq b Matpd, Kq. B bF ρpDq is an F1 linear space. Its K linear span is A1. Starting with the matrix A, get a matrix ˜ A in B bF ρpDq of the same rank, so rank is at least pn ´ 1qd ` 1. All matrices in B bF ρpDq have rank nd (over K) so ˜ A has rank nd

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

Proof of main lemma .....

Let ρ : D Ñ Matpd, Kq, be a representation of D. A1 :“ A b Matpd, Kq. Then A1 “ B b Matpd, Kq is a K-linear subspace of Matpn, Kq b Matpd, Kq. B bF ρpDq is an F1 linear space. Its K linear span is A1. Starting with the matrix A, get a matrix ˜ A in B bF ρpDq of the same rank, so rank is at least pn ´ 1qd ` 1. All matrices in B bF ρpDq have rank nd (over K) so ˜ A has rank nd Because F ě 2nd, we can find a matrix in A of rank nd using ideas from [dGIR96].

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

Proof of main lemma .....

Let ρ : D Ñ Matpd, Kq, be a representation of D. A1 :“ A b Matpd, Kq. Then A1 “ B b Matpd, Kq is a K-linear subspace of Matpn, Kq b Matpd, Kq. B bF ρpDq is an F1 linear space. Its K linear span is A1. Starting with the matrix A, get a matrix ˜ A in B bF ρpDq of the same rank, so rank is at least pn ´ 1qd ` 1. All matrices in B bF ρpDq have rank nd (over K) so ˜ A has rank nd Because F ě 2nd, we can find a matrix in A of rank nd using ideas from [dGIR96]. We need to construct division algebras, and be able to compute with them, at each stage

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

Using extension fields [dGIR96].

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

Using extension fields [dGIR96].

Assume K is an extension of F and you have a matrix in B b Matpd, Kq of rank r. Let S Ă F of size at least r.

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

Using extension fields [dGIR96].

Assume K is an extension of F and you have a matrix in B b Matpd, Kq of rank r. Let S Ă F of size at least r. Let B1, . . . , Bl be a F basis of B. Then A “ a1

1B1 ` a1 2B2 ` . . . ` a1 lBl, and there is a r ˆ r window

in A with nonzero determinant, say the principal r window.

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

Using extension fields [dGIR96].

Assume K is an extension of F and you have a matrix in B b Matpd, Kq of rank r. Let S Ă F of size at least r. Let B1, . . . , Bl be a F basis of B. Then A “ a1

1B1 ` a1 2B2 ` . . . ` a1 lBl, and there is a r ˆ r window

in A with nonzero determinant, say the principal r window. As a polynomial in x, the determinant of the principal r window xB1 ` a1

2B2 ` . . . ` a1 lBl is non zero. This is of

degree r. Since S has more than r elements there is an a1 P S Ă F such that the determinant a1B1 ` a1

2B2 ` . . . ` a1 lBl is non zero.

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

Using extension fields [dGIR96].

Assume K is an extension of F and you have a matrix in B b Matpd, Kq of rank r. Let S Ă F of size at least r. Let B1, . . . , Bl be a F basis of B. Then A “ a1

1B1 ` a1 2B2 ` . . . ` a1 lBl, and there is a r ˆ r window

in A with nonzero determinant, say the principal r window. As a polynomial in x, the determinant of the principal r window xB1 ` a1

2B2 ` . . . ` a1 lBl is non zero. This is of

degree r. Since S has more than r elements there is an a1 P S Ă F such that the determinant a1B1 ` a1

2B2 ` . . . ` a1 lBl is non zero.

Complete the proof by recursion, substituting values for a1

2, a1 3, . . . , a1 l.

Background and problem statement Using Gurvits algorithm Progress via Blow-ups Matrix of maximum rank

Second Wong sequence [IKQS14]

Background and problem statement Using Gurvits algorithm Progress via Blow-ups Matrix of maximum rank

Second Wong sequence [IKQS14]

Definition Given pA, Bq, A P Matpn, Fq and B ď Matpn, Fq, the second Wong sequence of pA, Bq is the following sequence of subspaces in Fn: W0 “ 0, W1 “ BpA´1pW0qq, . . . , Wi “ BpA´1pWi´1qq, . . . .

Background and problem statement Using Gurvits algorithm Progress via Blow-ups Matrix of maximum rank

Second Wong sequence [IKQS14]

Definition Given pA, Bq, A P Matpn, Fq and B ď Matpn, Fq, the second Wong sequence of pA, Bq is the following sequence of subspaces in Fn: W0 “ 0, W1 “ BpA´1pW0qq, . . . , Wi “ BpA´1pWi´1qq, . . . . W0 ă W1 ă W2 ă ¨ ¨ ¨ ă Wℓ “ Wℓ`1 “ . . . for some ℓ P t0, 1, . . . , nu. Wℓ is then called the limit of this sequence, denoted as W ˚.

Background and problem statement Using Gurvits algorithm Progress via Blow-ups Matrix of maximum rank

Second Wong sequence [IKQS14]

Definition Given pA, Bq, A P Matpn, Fq and B ď Matpn, Fq, the second Wong sequence of pA, Bq is the following sequence of subspaces in Fn: W0 “ 0, W1 “ BpA´1pW0qq, . . . , Wi “ BpA´1pWi´1qq, . . . . W0 ă W1 ă W2 ă ¨ ¨ ¨ ă Wℓ “ Wℓ`1 “ . . . for some ℓ P t0, 1, . . . , nu. Wℓ is then called the limit of this sequence, denoted as W ˚. When A P B, W ˚ ď impAq if and only if there exists a corankpAq-shrunk subspace

Background and problem statement Using Gurvits algorithm Progress via Blow-ups Matrix of maximum rank

Second Wong sequence [IKQS14]

Definition Given pA, Bq, A P Matpn, Fq and B ď Matpn, Fq, the second Wong sequence of pA, Bq is the following sequence of subspaces in Fn: W0 “ 0, W1 “ BpA´1pW0qq, . . . , Wi “ BpA´1pWi´1qq, . . . . W0 ă W1 ă W2 ă ¨ ¨ ¨ ă Wℓ “ Wℓ`1 “ . . . for some ℓ P t0, 1, . . . , nu. Wℓ is then called the limit of this sequence, denoted as W ˚. When A P B, W ˚ ď impAq if and only if there exists a corankpAq-shrunk subspace A is of maximum rank and A´1pW ˚q is a corank(A)-shrunk subspace.

Background and problem statement Using Gurvits algorithm Progress via Blow-ups Matrix of maximum rank

Using the second Wong sequence

Background and problem statement Using Gurvits algorithm Progress via Blow-ups Matrix of maximum rank

Using the second Wong sequence

What if A is not of maximum rank in Btd,du?

Background and problem statement Using Gurvits algorithm Progress via Blow-ups Matrix of maximum rank

Using the second Wong sequence

What if A is not of maximum rank in Btd,du? Incrementing rank Let B ď Matpn, Fq and let A “ Btd,du. Assume that we are given a matrix A P A with rkpAq “ rd, and |F| is Ωpndd1q, where d1 ą r is any positive integer. There exists a deterministic algorithm that returns either an pn ´ rqd-shrunk subspace for A (equivalently, an pn ´ rq-shrunk subspace for B), or a matrix B P A b Matpd1, Fq of rank at least pr ` 1qdd1.

Background and problem statement Using Gurvits algorithm Progress via Blow-ups Division algebras

Cyclic algebras and the construction of Dickson

Let K{F be a Galois extension with cyclic Galois group. Let σ be a generator of the Galois group and s “ dimFpKq. Take f P F and a symbol x, and consider D “ K ‘ K ¨ x ‘ K ¨ x2 ` . . . K ¨ xs´1. Multiply elements in D using the distributive law and using xs “ f and x ¨ b “ σpbqx for all b P K. F i in the center of D and so D is an F-algebra. Dimension

• ver F is s2.

Wedderburn - if f, f 2, . . . , f s´1 are not in NormpKq, then D is a division algebra, and in this case D bF K – MatpKq.

Determining the shrunk subspaces

Blowing-down a shrunk subspace

Determining the shrunk subspaces

Blowing-down a shrunk subspace

Shrinking by a factor of d If A “ Btd,du has an s-shrunk subspace, then A has an s1-shrunk subspace with s1 ě s and s.t. d divides s1. B has an s1{d-shrunk subspace.

Determining the shrunk subspaces

Blowing-down a shrunk subspace

Shrinking by a factor of d If A “ Btd,du has an s-shrunk subspace, then A has an s1-shrunk subspace with s1 ě s and s.t. d divides s1. B has an s1{d-shrunk subspace. Idea Maximal shrunk subspaces are of the form Uo b Fd and their image under A is of the form Wo b Fd.

Determining the shrunk subspaces

Blowing-down

Determining the shrunk subspaces

Blowing-down

Reducing the size of blow-ups Let B ď Matpn, Fq, and d ą n ` 1. Assume we are given a matrix A P Btd,du of rank dn. Then there exists a deterministic polynomial-time procedure that constructs A1 P Btd´1,d´1u of rank pd ´ 1qn.

Determining the shrunk subspaces

Construction of division algebras Let L be a cyclic extension of degree d of a field K 1. Let σ be a generator of the Galois group. Consider the transcendental extension LpZq of L. Then σ extends to an automorphism (denoted again by σ) of LpZq such that the fixed field of σ is K 1pZq. Thus LpZq is a cyclic extension of K 1pZq. Consider the K 1pZq-algebra D generated by (a basis for) L and by an element U with relations Ud “ Z and Ua “ aσU (@a P LpZq, or, equivalently @a P the basis for L). Then D is a central division algebra of index d over K 1pZq.

Determining the shrunk subspaces

Open problems

Determining the shrunk subspaces

Open problems

Get a combinatorial algorithm in characteristic zero.

Determining the shrunk subspaces

Open problems

Get a combinatorial algorithm in characteristic zero. Is there an augmenting path algorithm?

Determining the shrunk subspaces

Open problems

Get a combinatorial algorithm in characteristic zero. Is there an augmenting path algorithm? For the GCT programme, desingularizing the null cone may be important - this may help isolate points which are in the border.

Determining the shrunk subspaces

Open problems

Get a combinatorial algorithm in characteristic zero. Is there an augmenting path algorithm? For the GCT programme, desingularizing the null cone may be important - this may help isolate points which are in the border. Orbit closure problem for the left right action

Determining the shrunk subspaces

Open problems

Get a combinatorial algorithm in characteristic zero. Is there an augmenting path algorithm? For the GCT programme, desingularizing the null cone may be important - this may help isolate points which are in the border. Orbit closure problem for the left right action .. NNL for this invariant ring.

Determining the shrunk subspaces

References I

• B. Adsul, S. Nayak, and K. V. Subrahmanyam.

A geometric approach to the Kronecker problem II: rectangular shapes, invariants of matrices and the Artin–Procesi theorem. preprint, 2007.

• M. Bürgin and J. Draisma.

The Hilbert null-cone on tuples of matrices and bilinear forms. Mathematische Zeitschrift, 254(4):785–809, 2006. Harm Derksen. Polynomial bounds for rings of invariants. Proceedings of the American Mathematical Society, 129(4):955–964, 2001.

Determining the shrunk subspaces

References II

Willem A. de Graaf, Gábor Ivanyos, and Lajos Rónyai. Computing Cartan subalgebras of Lie algebras. Applicable Algebra in Engineering, Communication and Computing, 7(5):339–349, 1996. Harm Derksen and Visu Makam. Polynomial degree bounds for matrix semi-invariants. preprint, 2015.

• M. Domokos and A. N. Zubkov.

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