On Strassen s Conjecture

Transcription

1 On Strassen s Conjecture Elisa Postinghel (KU Leuven) joint with Jarek Buczyński (IMPAN/MIMUW) Daejeon August 3-7, 2015 Elisa Postinghel (KU Leuven) () On Strassen s Conjecture SIAM AG / 13

2 Introduction: matrix multiplication Multiplication of two n n matrices: M n,n,n : C n2 C n2 C n2 M 1, M 2 M 1 M 2 Standard algorithm: O(n 3 ) arithmetic operations (n 3 multiplications & n 3 n 2 additions) Elisa Postinghel (KU Leuven) () On Strassen s Conjecture SIAM AG / 13

3 Introduction: matrix multiplication Multiplication of two n n matrices: M n,n,n : C n2 C n2 C n2 M 1, M 2 M 1 M 2 Standard algorithm: O(n 3 ) arithmetic operations (n 3 multiplications & n 3 n 2 additions) [Strassen s algorithm, 69] n = 2: needs at most 7 multiplications (instead of 2 3 = 8) [Winograd, 71] Cannot be better than 7 for 2 2 matrices Elisa Postinghel (KU Leuven) () On Strassen s Conjecture SIAM AG / 13

4 Introduction: matrix multiplication Multiplication of two n n matrices: M n,n,n : C n2 C n2 C n2 M 1, M 2 M 1 M 2 Standard algorithm: O(n 3 ) arithmetic operations (n 3 multiplications & n 3 n 2 additions) [Strassen s algorithm, 69] n = 2: needs at most 7 multiplications (instead of 2 3 = 8) [Winograd, 71] Cannot be better than 7 for 2 2 matrices Generalisation of Strassen s algorithm: needs n log 2 (7) = n 2.81 Elisa Postinghel (KU Leuven) () On Strassen s Conjecture SIAM AG / 13

5 Matrix multiplication as a tensor V 1, V 2, V 3 vector spaces of dimensions n 1, n 2, n 3 {n i n j matrices} {linear maps V i V j } {elements in V i V j } Elisa Postinghel (KU Leuven) () On Strassen s Conjecture SIAM AG / 13

6 Matrix multiplication as a tensor V 1, V 2, V 3 vector spaces of dimensions n 1, n 2, n 3 {n i n j matrices} {linear maps V i V j } {elements in V i V j } M n1,n 2,n 3 : (V 1 V 2) (V 2 V 3) (V 1 V 3) (v 1 v 2) (v 2 v 3) v 2 (v 2)v 1 v 3 Elisa Postinghel (KU Leuven) () On Strassen s Conjecture SIAM AG / 13

7 Matrix multiplication as a tensor V 1, V 2, V 3 vector spaces of dimensions n 1, n 2, n 3 {n i n j matrices} {linear maps V i V j } {elements in V i V j } i.e. M n1,n 2,n 3 : (V 1 V 2) (V 2 V 3) (V 1 V 3) (v 1 v 2) (v 2 v 3) v 2 (v 2)v 1 v 3 M n1,n 2,n 3 = id V1 id V2 id V3 ((V 1 V 2 ) (V 2 V 3 )) (V 1 V 3 ) = (V 1 V 2 ) (V 2 V 3 ) (V 1 V 3 ) Elisa Postinghel (KU Leuven) () On Strassen s Conjecture SIAM AG / 13

8 Tensor rank Consider the bilinear map T : A B C Elisa Postinghel (KU Leuven) () On Strassen s Conjecture SIAM AG / 13

9 Tensor rank Consider the bilinear map T : A B C The simplest bilinear maps are: take α A, β B and c C: α β c : (a, b) A B α(a)β(b)c C Elisa Postinghel (KU Leuven) () On Strassen s Conjecture SIAM AG / 13

10 Tensor rank Consider the bilinear map T : A B C The simplest bilinear maps are: take α A, β B and c C: α β c : (a, b) A B α(a)β(b)c C Every tensor can be represented as T (a, b) = r α i (a)β i (b)c i i=1 Definition The rank of T, R(T ), is the minimal number over all such presentations. Elisa Postinghel (KU Leuven) () On Strassen s Conjecture SIAM AG / 13

11 Tensor rank Consider the bilinear map T : A B C The simplest bilinear maps are: take α A, β B and c C: α β c : (a, b) A B α(a)β(b)c C Every tensor can be represented as T (a, b) = r α i (a)β i (b)c i i=1 Definition The rank of T, R(T ), is the minimal number over all such presentations. In particular, the rank R(M n1,n 2,n 3 ) measures the number of multiplications needed to compute the matrix multiplication. Elisa Postinghel (KU Leuven) () On Strassen s Conjecture SIAM AG / 13

12 Strassen s additivity conjecture Consider A = A 1 A 2, B = B 1 B 2, C = C 1 C 2 A 1 = C a 1 etc. Elisa Postinghel (KU Leuven) () On Strassen s Conjecture SIAM AG / 13

13 Strassen s additivity conjecture Consider A = A 1 A 2, B = B 1 B 2, C = C 1 C 2 A 1 = C a 1 etc. Take T 1 A 1 B 1 C 1, T 2 A 2 B 2 C 2 Elisa Postinghel (KU Leuven) () On Strassen s Conjecture SIAM AG / 13

14 Strassen s additivity conjecture Consider A = A 1 A 2, B = B 1 B 2, C = C 1 C 2 A 1 = C a 1 etc. Take T 1 A 1 B 1 C 1, T 2 A 2 B 2 C 2 Conjecture (Strassen s Additivity Conjecture (SAC), 73) Then R(T 1 + T 2 ) = R(T 1 ) + R(T 2 ). Elisa Postinghel (KU Leuven) () On Strassen s Conjecture SIAM AG / 13

15 Strassen s additivity conjecture Consider A = A 1 A 2, B = B 1 B 2, C = C 1 C 2 A 1 = C a 1 etc. Take T 1 A 1 B 1 C 1, T 2 A 2 B 2 C 2 Conjecture (Strassen s Additivity Conjecture (SAC), 73) Then R(T 1 + T 2 ) = R(T 1 ) + R(T 2 ). Theorem (Buczyński-Landsberg, 13) If R(T 1 ) max{a 1, b 1, c 1 }, then SAC holds. Corollary If a 1 b 1 c 1, then SAC holds. If c 1 = 1, then SAC holds. Elisa Postinghel (KU Leuven) () On Strassen s Conjecture SIAM AG / 13

16 SAC holds if one of the dimensions involved is 2 Theorem (Ja Ja-Takche 86; Buczyński-P) If 2 {a 1, a 2, b 1, b 2, c 1, c 2 } then SAC holds. Elisa Postinghel (KU Leuven) () On Strassen s Conjecture SIAM AG / 13

17 SAC holds if one of the dimensions involved is 2 Theorem (Ja Ja-Takche 86; Buczyński-P) If 2 {a 1, a 2, b 1, b 2, c 1, c 2 } then SAC holds. Original proof by Ja Ja and Takche relies on some non-trivial facts from linear algebra, such as Kronecker s theory of pencils, and on the theory of invariant polynomials. Our proof is simpler and uses geometry. Lemma If b 1 c 1 > a 1 b 1 c 1 b 1 c 1 + 2, then there exists a rank-1 tensor in the image of T 1 : A 1 B 1 C 1. Elisa Postinghel (KU Leuven) () On Strassen s Conjecture SIAM AG / 13

18 SAC holds if one of the dimensions involved is 2 Theorem (Ja Ja-Takche 86; Buczyński-P) If 2 {a 1, a 2, b 1, b 2, c 1, c 2 } then SAC holds. Original proof by Ja Ja and Takche relies on some non-trivial facts from linear algebra, such as Kronecker s theory of pencils, and on the theory of invariant polynomials. Our proof is simpler and uses geometry. Lemma If b 1 c 1 > a 1 b 1 c 1 b 1 c 1 + 2, then there exists a rank-1 tensor in the image of T 1 : A 1 B 1 C 1. Proof. dim P(T 1 (A 1 )) = a 1 1, Elisa Postinghel (KU Leuven) () On Strassen s Conjecture SIAM AG / 13

19 SAC holds if one of the dimensions involved is 2 Theorem (Ja Ja-Takche 86; Buczyński-P) If 2 {a 1, a 2, b 1, b 2, c 1, c 2 } then SAC holds. Original proof by Ja Ja and Takche relies on some non-trivial facts from linear algebra, such as Kronecker s theory of pencils, and on the theory of invariant polynomials. Our proof is simpler and uses geometry. Lemma If b 1 c 1 > a 1 b 1 c 1 b 1 c 1 + 2, then there exists a rank-1 tensor in the image of T 1 : A 1 B 1 C 1. Proof. dim P(T 1 (A 1 )) = a 1 1, codim Seg(P(B 1 ) P(C 1 )) = b 1 c 1 b 1 c Elisa Postinghel (KU Leuven) () On Strassen s Conjecture SIAM AG / 13

20 SAC holds if one of the dimensions involved is 2 Theorem (Ja Ja-Takche 86; Buczyński-P) If 2 {a 1, a 2, b 1, b 2, c 1, c 2 } then SAC holds. Original proof by Ja Ja and Takche relies on some non-trivial facts from linear algebra, such as Kronecker s theory of pencils, and on the theory of invariant polynomials. Our proof is simpler and uses geometry. Lemma If b 1 c 1 > a 1 b 1 c 1 b 1 c 1 + 2, then there exists a rank-1 tensor in the image of T 1 : A 1 B 1 C 1. Proof. dim P(T 1 (A 1 )) = a 1 1, codim Seg(P(B 1 ) P(C 1 )) = b 1 c 1 b 1 c Hence the varieties P(T 1 (A 1 )) and Seg(P(B 1) P(C 1 )) have non-empty intersection in P(B 1 C 1 ). Elisa Postinghel (KU Leuven) () On Strassen s Conjecture SIAM AG / 13

21 SAC holds if one of the dimensions involved is 2 Proof of Thm [Buczyński-P]: Assume T 1, T 2 concise and assume, w.l.o.g, that c 1 = 2. Elisa Postinghel (KU Leuven) () On Strassen s Conjecture SIAM AG / 13

22 SAC holds if one of the dimensions involved is 2 Proof of Thm [Buczyński-P]: Assume T 1, T 2 concise and assume, w.l.o.g, that c 1 = 2. Assume that T = T 1 + T 2 is minimal (lowest rank) such that SAC does not hold, namely such that R(T ) < R(T 1 ) + R(T 2 ). Elisa Postinghel (KU Leuven) () On Strassen s Conjecture SIAM AG / 13

23 SAC holds if one of the dimensions involved is 2 Proof of Thm [Buczyński-P]: Assume T 1, T 2 concise and assume, w.l.o.g, that c 1 = 2. Assume that T = T 1 + T 2 is minimal (lowest rank) such that SAC does not hold, namely such that R(T ) < R(T 1 ) + R(T 2 ). 1 By Lemma, there exists a rank-1 tensor in the image of T 1 (A 1 ) B 1 C 1, and thus also in the image T ((A 1 A 2 ) ) B C. 2 Hence there exists a minimal decomposition of T 1 + T 2 in rank-1 tensors such that one summand, q is an element in A 1 B 1 C 1. Elisa Postinghel (KU Leuven) () On Strassen s Conjecture SIAM AG / 13

24 SAC holds if one of the dimensions involved is 2 Proof of Thm [Buczyński-P]: Assume T 1, T 2 concise and assume, w.l.o.g, that c 1 = 2. Assume that T = T 1 + T 2 is minimal (lowest rank) such that SAC does not hold, namely such that R(T ) < R(T 1 ) + R(T 2 ). 1 By Lemma, there exists a rank-1 tensor in the image of T 1 (A 1 ) B 1 C 1, and thus also in the image T ((A 1 A 2 ) ) B C. 2 Hence there exists a minimal decomposition of T 1 + T 2 in rank-1 tensors such that one summand, q is an element in A 1 B 1 C 1. 3 We can reduce to T q = (T 1 q) + T 2, with T 1 q A 1 B 1 C 1 and R(T q) < R(T 1 q) + R(T 2 ) and find a smaller counterexample. Contradiction. Elisa Postinghel (KU Leuven) () On Strassen s Conjecture SIAM AG / 13

25 Symmetric Strassen conjecture Let F be a degree-d form in C[x 1,..., x n ]. Definition The symmetric rank of F, R s (F ), is the minimal r such that F = d i=1 Ld i, Elisa Postinghel (KU Leuven) () On Strassen s Conjecture SIAM AG / 13

26 Symmetric Strassen conjecture Let F be a degree-d form in C[x 1,..., x n ]. Definition The symmetric rank of F, R s (F ), is the minimal r such that F = d i=1 Ld i, Conjecture (Symmetric SAC) If F i C[x 1,..., x ni ] d, i = 1, 2, then R s (F + G) = R s (F ) + R s (G). Elisa Postinghel (KU Leuven) () On Strassen s Conjecture SIAM AG / 13

27 Symmetric Strassen conjecture Let F be a degree-d form in C[x 1,..., x n ]. Definition The symmetric rank of F, R s (F ), is the minimal r such that F = d i=1 Ld i, Conjecture (Symmetric SAC) If F i C[x 1,..., x ni ] d, i = 1, 2, then R s (F + G) = R s (F ) + R s (G). Theorem (Carlini-Catalisano-Chiantini 15) True for binary forms (n 1, n 2 = 2). Theorem (Carlini-Catalisano-Chiantini-Geramita-Woo 15) True for arbitrary sums of binary forms. Proof. Uses Hilbert functions of sets of points and apolarity. Elisa Postinghel (KU Leuven) () On Strassen s Conjecture SIAM AG / 13

28 Symmetric Strassen conjecture Let F be a degree-d form in C[x 1,..., x n ]. Definition The symmetric rank of F, R s (F ), is the minimal r such that F = d i=1 Ld i, Conjecture (Symmetric SAC) If F i C[x 1,..., x ni ] d, i = 1, 2, then R s (F + G) = R s (F ) + R s (G). Theorem (Carlini-Catalisano-Chiantini 15) True for binary forms (n 1, n 2 = 2). Theorem (Carlini-Catalisano-Chiantini-Geramita-Woo 15) True for arbitrary sums of binary forms. Proof. Uses Hilbert functions of sets of points and apolarity. Conjecture (Comon s conjecture) R s (F ) = R(F ). Elisa Postinghel (KU Leuven) () On Strassen s Conjecture SIAM AG / 13

29 Additivity for border rank of tensors Definition A tensor T A B C has border rank r = R(T ) if it is a limit of tensors of rank r, but not a limit of tensors of rank < r. {T : R(T ) r} σ r (Seg(P(A) P(B) P(C)) P(A B C) Elisa Postinghel (KU Leuven) () On Strassen s Conjecture SIAM AG / 13

30 Additivity for border rank of tensors Definition A tensor T A B C has border rank r = R(T ) if it is a limit of tensors of rank r, but not a limit of tensors of rank < r. {T : R(T ) r} σ r (Seg(P(A) P(B) P(C)) P(A B C) Example (Landsberg 06) R(M 2,2,2 ) = R(M 2,2,2 ) = 7 Elisa Postinghel (KU Leuven) () On Strassen s Conjecture SIAM AG / 13

31 Additivity for border rank of tensors Definition A tensor T A B C has border rank r = R(T ) if it is a limit of tensors of rank r, but not a limit of tensors of rank < r. {T : R(T ) r} σ r (Seg(P(A) P(B) P(C)) P(A B C) Example (Landsberg 06) R(M 2,2,2 ) = R(M 2,2,2 ) = 7 Question (Border SAC) R(T 1 + T 2 ) = R(T 1 ) + R(T 1 )? Elisa Postinghel (KU Leuven) () On Strassen s Conjecture SIAM AG / 13

32 Additivity for border rank of tensors Definition A tensor T A B C has border rank r = R(T ) if it is a limit of tensors of rank r, but not a limit of tensors of rank < r. {T : R(T ) r} σ r (Seg(P(A) P(B) P(C)) P(A B C) Example (Landsberg 06) R(M 2,2,2 ) = R(M 2,2,2 ) = 7 Question (Border SAC) R(T 1 + T 2 ) = R(T 1 ) + R(T 1 )? Counterexample (Schönhage 81) Border SAC is false if (a 1 + a 2, b 1 + b 2, c 1 + c 2 ) = (6 + 1, 3 + 2, 2 + 2). Elisa Postinghel (KU Leuven) () On Strassen s Conjecture SIAM AG / 13

33 Additivity for border rank of tensors? Yes Case (a 1 + a 2, b 1 + b 2, c 1 + c 2 ) = (3 + 1, 3 + 1, 3 + 1) with R(T 1 ) = 5. Elisa Postinghel (KU Leuven) () On Strassen s Conjecture SIAM AG / 13

34 Additivity for border rank of tensors? Yes Case (a 1 + a 2, b 1 + b 2, c 1 + c 2 ) = (3 + 1, 3 + 1, 3 + 1) with R(T 1 ) = 5. 3 T 1 = a i X i (assume concise) i=1 {a 1, a 2, a 3 } basis for A 1 = C 3 X i B 1 C 1 = C 3 C 3 Elisa Postinghel (KU Leuven) () On Strassen s Conjecture SIAM AG / 13

35 Additivity for border rank of tensors? Yes Case (a 1 + a 2, b 1 + b 2, c 1 + c 2 ) = (3 + 1, 3 + 1, 3 + 1) with R(T 1 ) = 5. T 1 = 3 a i X i i=1 (assume concise) {a 1, a 2, a 3 } basis for A 1 = C 3 X i B 1 C 1 = C 3 C 3 Strassen s degree-9 equation for σ 4 (Seg(P 2 P 2 P 2 )) 0 X 3 X 2 det(m(x 1, X 2, X 3 )) = X 3 0 X 1 X 2 X 1 0 = 0 Elisa Postinghel (KU Leuven) () On Strassen s Conjecture SIAM AG / 13

36 Additivity for border rank of tensors? Yes Case (a 1 + a 2, b 1 + b 2, c 1 + c 2 ) = (3 + 1, 3 + 1, 3 + 1) with R(T 1 ) = 5. T 1 = 3 a i X i i=1 (assume concise) {a 1, a 2, a 3 } basis for A 1 = C 3 X i B 1 C 1 = C 3 C 3 Strassen s degree-9 equation for σ 4 (Seg(P 2 P 2 P 2 )) 0 X 3 X 2 det(m(x 1, X 2, X 3 )) = X 3 0 X 1 X 2 X 1 0 = 0 T 2 = a 4 b c C C C Elisa Postinghel (KU Leuven) () On Strassen s Conjecture SIAM AG / 13

37 Additivity for border rank of tensors? Yes Case (a 1 + a 2, b 1 + b 2, c 1 + c 2 ) = (3 + 1, 3 + 1, 3 + 1) with R(T 1 ) = 5. T 1 = 3 a i X i i=1 (assume concise) {a 1, a 2, a 3 } basis for A 1 = C 3 X i B 1 C 1 = C 3 C 3 Strassen s degree-9 equation for σ 4 (Seg(P 2 P 2 P 2 )) 0 X 3 X 2 det(m(x 1, X 2, X 3 )) = X 3 0 X 1 X 2 X 1 0 = 0 with Y i = T 2 = a 4 b c C C C T 1 + T 2 = a 1 Y a 4 Y 4 ( Xi ) ( , i {1, 2, 3} and Y 4 = 0 1 ) Elisa Postinghel (KU Leuven) () On Strassen s Conjecture SIAM AG / 13

38 Additivity for border rank of tensors? Yes Y i = ( Xi ) ( , i {1, 2, 3} and Y 4 = 0 1 ) Elisa Postinghel (KU Leuven) () On Strassen s Conjecture SIAM AG / 13

39 Additivity for border rank of tensors? Yes Y i = ( Xi ) ( , i {1, 2, 3} and Y 4 = 0 1 Take the projection π : A B C A 1 B C a i a i, i {1, 2, 3} a 4 a 1 + a 2 + a 3. ) Elisa Postinghel (KU Leuven) () On Strassen s Conjecture SIAM AG / 13

40 Additivity for border rank of tensors? Yes Y i = ( Xi ) ( , i {1, 2, 3} and Y 4 = 0 1 Take the projection π : A B C A 1 B C a i a i, i {1, 2, 3} a 4 a 1 + a 2 + a 3. T 1 + T 2 3 i=1 a i Y i := 3 i=1 ) ( Xi 0 a i 0 1 ) Elisa Postinghel (KU Leuven) () On Strassen s Conjecture SIAM AG / 13

41 Additivity for border rank of tensors? Yes Y i = ( Xi ) ( , i {1, 2, 3} and Y 4 = 0 1 Take the projection π : A B C A 1 B C a i a i, i {1, 2, 3} a 4 a 1 + a 2 + a 3. T 1 + T 2 3 i=1 a i Y i := Strassen s equations for σ 5 (Seg(P 2 P 3 P 3 )) 10 rk M(Y 1, Y 2, Y 3) 3 i=1 ) ( Xi 0 a i 0 1 ) Elisa Postinghel (KU Leuven) () On Strassen s Conjecture SIAM AG / 13

42 Additivity for border rank of tensors? Yes Y i = ( Xi ) ( , i {1, 2, 3} and Y 4 = 0 1 Take the projection π : A B C A 1 B C a i a i, i {1, 2, 3} a 4 a 1 + a 2 + a 3. T 1 + T 2 3 i=1 a i Y i := 3 i=1 ) ( Xi 0 a i 0 1 Strassen s equations for σ 5 (Seg(P 2 P 3 P 3 )) ( ) 10 rk M(Y 1, Y 2, Y 3) M(X1, X = rk 2, X 3 ) 0 0 M(1, 1, 1) ) Elisa Postinghel (KU Leuven) () On Strassen s Conjecture SIAM AG / 13

43 Additivity for border rank of tensors? Yes Y i = ( Xi ) ( , i {1, 2, 3} and Y 4 = 0 1 Take the projection π : A B C A 1 B C a i a i, i {1, 2, 3} a 4 a 1 + a 2 + a 3. T 1 + T 2 3 i=1 a i Y i := 3 i=1 ) ( Xi 0 a i 0 1 Strassen s equations for σ 5 (Seg(P 2 P 3 P 3 )) ( ) 10 rk M(Y 1, Y 2, Y 3) M(X1, X = rk 2, X 3 ) 0 0 M(1, 1, 1) rk M(Y 1, Y 2, Y 3 ) = = 11 ) Elisa Postinghel (KU Leuven) () On Strassen s Conjecture SIAM AG / 13

44 Additivity for border rank of tensors? Yes Y i = ( Xi ) ( , i {1, 2, 3} and Y 4 = 0 1 Take the projection π : A B C A 1 B C a i a i, i {1, 2, 3} a 4 a 1 + a 2 + a 3. T 1 + T 2 3 i=1 a i Y i := 3 i=1 ) ( Xi 0 a i 0 1 Strassen s equations for σ 5 (Seg(P 2 P 3 P 3 )) ( ) 10 rk M(Y 1, Y 2, Y 3) M(X1, X = rk 2, X 3 ) 0 0 M(1, 1, 1) rk M(Y 1, Y 2, Y 3 ) = = 11 R(T 1 + T 2 ) 6 ) Elisa Postinghel (KU Leuven) () On Strassen s Conjecture SIAM AG / 13

45 Additivity for border rank of tensors? Yes Y i = ( Xi ) ( , i {1, 2, 3} and Y 4 = 0 1 Take the projection π : A B C A 1 B C a i a i, i {1, 2, 3} a 4 a 1 + a 2 + a 3. T 1 + T 2 3 i=1 a i Y i := 3 i=1 ) ( Xi 0 a i 0 1 Strassen s equations for σ 5 (Seg(P 2 P 3 P 3 )) ( ) 10 rk M(Y 1, Y 2, Y 3) M(X1, X = rk 2, X 3 ) 0 0 M(1, 1, 1) rk M(Y 1, Y 2, Y 3 ) = = 11 R(T 1 + T 2 ) 6 border SAC holds! Elisa Postinghel (KU Leuven) () On Strassen s Conjecture SIAM AG / 13 )

46 Additivity for border rank of tensors? Yes Using (variants of) Strassen s equations for σ 4 : border SAC holds if (a 1 + a 2, b 1 + b 2, c 1 + c 2 ) = (3 + 1) 3 with R(T 1 ) = 5, R(T 2 ) = 1. Elisa Postinghel (KU Leuven) () On Strassen s Conjecture SIAM AG / 13

47 Additivity for border rank of tensors? Yes Using (variants of) Strassen s equations for σ 4 : border SAC holds if (a 1 + a 2, b 1 + b 2, c 1 + c 2 ) = (3 + 1) 3 with R(T 1 ) = 5, R(T 2 ) = 1. (a 1 + a 2, b 1 + b 2, c 1 + c 2 ) = (3 + 2) 3 with R(T 1 ) = 5, R(T 2 ) = 2. (a 1 + a 2, b 1 + b 2, c 1 + c 2 ) = (3 + 1, (2 + 2) 2 ) with R(T 1 ) = 3,R(T 2 ) = 2. Elisa Postinghel (KU Leuven) () On Strassen s Conjecture SIAM AG / 13

48 Additivity for border rank of tensors? Yes Using (variants of) Strassen s equations for σ 4 : border SAC holds if (a 1 + a 2, b 1 + b 2, c 1 + c 2 ) = (3 + 1) 3 with R(T 1 ) = 5, R(T 2 ) = 1. (a 1 + a 2, b 1 + b 2, c 1 + c 2 ) = (3 + 2) 3 with R(T 1 ) = 5, R(T 2 ) = 2. (a 1 + a 2, b 1 + b 2, c 1 + c 2 ) = (3 + 1, (2 + 2) 2 ) with R(T 1 ) = 3,R(T 2 ) = 2. Similarly, using Friedland s equations for σ 3 (Seg(P 2 P 3 P 3 )): border SAC holds for (a 1 + a 2, b 1 + b 2, c 1 + c 2 ) = (3 + 1) 3 with R(T 1 ) = 4, R(T 2 ) = 1. Elisa Postinghel (KU Leuven) () On Strassen s Conjecture SIAM AG / 13

49 Bibliography [1] J. Buczyński, J. M. Landsberg Ranks of tensors and a generalization of secant varieties, Linear Algebra Appl. 438 (2013), no. 2, A69 [2] J. Buczyński, E. Postinghel, Remarks on Strassen s conjecture, In preparation. [3] E. Carlini, M. V. Catalisano, L. Chiantini Progress on the symmetric Strassen conjecture, J. Pure Appl. Algebra 219 (2015), no. 8, [4] J. Ja Ja, J. Takche, On the validity of the direct sum conjecture, SIAM J. Comput. 15 (1986), no. 4, [5] J. M. Landsberg, The border rank of the multiplication of 22 matrices is seven, J. Amer. Math. Soc. 19 (2006), no. 2, [6] A. Schönhage, Partial and total matrix multiplication, SIAM J. Comput. 10 (1981), no. 3, [7] V. Strassen, Gaussian elimination is not optimal, Numer. Math. 13 (1969) [8] V. Strassen, Vermeidung von Divisionen, J. Reine Angew. Math. 264 (1973), THANKS!! Elisa Postinghel (KU Leuven) () On Strassen s Conjecture SIAM AG / 13