Ranks and generalized ranks

Transcription

1 Ranks and generalized ranks Zach Teitler Boise State University SIAM Conference on Applied Algebraic Geometry Minisymposium on Algebraic Geometry of Tensor Decompositions and its Applications October 7, 2011

2 Zach Teitler (BSU) Ranks and generalized ranks 2 / 10 1 Background 2 [Landsberg-T.] Bound for ranks of polynomials 3 Multihomogeneous polynomials 4 Work in progress: Young flattenings

3 Zach Teitler (BSU) Ranks and generalized ranks 3 / 10 φ S d W = C[x 0,..., x n ] d homogeneous polynomial of degree d φ = c 1 l d c r l d r power sum decomposition (Waring) rank R(φ) = least r in a power sum decomposition Problem: Find R(φ). xy = 1 4 ((x + y) 2 (x y) 2) R(xy) = 2

4 Zach Teitler (BSU) Ranks and generalized ranks 3 / 10 φ S d W = C[x 0,..., x n ] d homogeneous polynomial of degree d φ = c 1 l d c r l d r power sum decomposition (Waring) rank R(φ) = least r in a power sum decomposition Problem: Find R(φ). xyz = 1 ( (x + y + z) 3 (x + y z) 3 (x y + z) 3 + (x y z) 3), 24 R(xyz) 4

5 Zach Teitler (BSU) Ranks and generalized ranks 3 / 10 φ S d W = C[x 0,..., x n ] d homogeneous polynomial of degree d φ = c 1 l d c r l d r power sum decomposition (Waring) rank R(φ) = least r in a power sum decomposition Problem: Find R(φ). x 1 x n = 1 ±(x1 2 n 1 ± x 2 ± ± x n ) n n! R(x 1 x n ) 2 n 1

6 Zach Teitler (BSU) Ranks and generalized ranks 4 / 10 R(φ) = least r such that φ = c 1 l d c r l d r Ranks are known: deg φ = 2 (R(φ) = rank of symmetric matrix) n = 1 (Sylvester, 1851) φ S d W general (Alexander Hirschowitz, 1995) Plane cubic curves Some natural questions: R(x 1 x n ) = 2 n 1? Monomials: R(x b 1 1 x bn n ) (b 2 + 1) (b n + 1). Equality? What is R(det n ) (n n determinant of (x i,j ) 1 i,j n )? 14 R(det 3 ) 24 (Landsberg-T.) (improved from 9 R(det 3 ) 24, Sylvester s catalecticant bound)

7 Zach Teitler (BSU) Ranks and generalized ranks 4 / 10 R(φ) = least r such that φ = c 1 l d c r l d r Ranks are known: deg φ = 2 (R(φ) = rank of symmetric matrix) n = 1 (Sylvester, 1851) φ S d W general (Alexander Hirschowitz, 1995) Plane cubic curves Some natural questions: R(x 1 x n ) = 2 n 1? R( (x 1 x n ) d ) = (d + 1) n 1 (Ranestad Schreyer, April 2011) Monomials: R(x b 1 1 x bn n ) (b 2 + 1) (b n + 1). Equality? What is R(det n ) (n n determinant of (x i,j ) 1 i,j n )? 14 R(det 3 ) 24 (Landsberg-T.) (improved from 9 R(det 3 ) 24, Sylvester s catalecticant bound)

8 Zach Teitler (BSU) Ranks and generalized ranks 4 / 10 R(φ) = least r such that φ = c 1 l d c r l d r Ranks are known: deg φ = 2 (R(φ) = rank of symmetric matrix) n = 1 (Sylvester, 1851) φ S d W general (Alexander Hirschowitz, 1995) Plane cubic curves Some natural questions: R(x 1 x n ) = 2 n 1? R( (x 1 x n ) d ) = (d + 1) n 1 (Ranestad Schreyer, April 2011) Monomials: R(x b 1 1 x n bn ) (b 2 + 1) (b n + 1). Equality? Yes. (Carlini-Catalisano-Geramita, this Wednesday, Oct. 4, 2011) What is R(det n ) (n n determinant of (x i,j ) 1 i,j n )? 14 R(det 3 ) 24 (Landsberg-T.) (improved from 9 R(det 3 ) 24, Sylvester s catalecticant bound)

9 Zach Teitler (BSU) Ranks and generalized ranks 4 / 10 R(φ) = least r such that φ = c 1 l d c r l d r Ranks are known: deg φ = 2 (R(φ) = rank of symmetric matrix) n = 1 (Sylvester, 1851) φ S d W general (Alexander Hirschowitz, 1995) Plane cubic curves Some natural questions: R(x 1 x n ) = 2 n 1? R( (x 1 x n ) d ) = (d + 1) n 1 (Ranestad Schreyer, April 2011) Monomials: R(x b 1 1 x n bn ) (b 2 + 1) (b n + 1). Equality? Yes. (Carlini-Catalisano-Geramita, this Wednesday, Oct. 4, 2011) What is R(det n ) (n n determinant of (x i,j ) 1 i,j n )? 14 R(det 3 ) 24 (Landsberg-T.) (improved from 9 R(det 3 ) 24, Sylvester s catalecticant bound) Still wide open.

10 Zach Teitler (BSU) Ranks and generalized ranks 5 / 10 φ S d W, 0 a d: a th catalecticant Sylvester: R(φ) rank φ a 4 R(xyz) rank(xyz) 1 = 3. φ a : S d a W S a W, D D(φ)

11 Zach Teitler (BSU) Ranks and generalized ranks 6 / 10 φ a : S d a W S a W Theorem (Landsberg-T.) Let φ S d W, 0 a d. Assume φ S d V, V W = V = W (equiv.: V (φ) not a cone; φ 1 surjective). Let Σ a = {p : mult p (φ) > a} = V (a th derivatives of φ). Then R(φ) rank φ a + dim Σ a R(xyz) rank(xyz) 1 + dim Sing V (xyz) + 1 = = R(det 3 ) 14 (thm above) 9 (catalecticant) 8 R(xyzw) 8 (ad hoc) 7 (thm) 6 (catalecticant) Proof: Count dimensions of subvarieties in P ker φ a.

12 Zach Teitler (BSU) Ranks and generalized ranks 7 / 10 Generalize to multihomogeneous polynomials: Homogeneous polynomial: rank with respect to Veronese Multihomogeneous Segre-Veronese Still have catalecticant bound ( symmetric flattening ) Try to add dimension of some set of singularities

13 Zach Teitler (BSU) Ranks and generalized ranks 8 / 10 Theorem Let M S d i W i, a = (a 1,..., a k ). Ma : S d i a i Wi S a i W i. Then R(φ) rank Ma (well-known). Suppose M a is surjective, a i = 1 if a i 0, 0 otherwise. Let Σa = {p : mult p (M) > a} = V (a th derivatives of M). Then R(M) rank Ma + dim Σa + 1. Condition M a surjective analogous to condition φ 1 surjective Geometric version of this condition? Σ a = V (image of φ a or Ma)

14 Zach Teitler (BSU) Ranks and generalized ranks 9 / 10 Work in progress: Young flattenings (Landsberg Ottaviani, 2010) L very ample line bundle on X, v PH 0 (L) X ; find R(v) = R X (v) E any vector bundle, E E L L H 0 (E) H 0 (L E ) H 0 (L) v C Cv E : H 0 (L E ) H 0 (E), generalized catalecticant Previous results: L = O(d), E = O(a); similar for multihomogeneous. [L-O] R(v) rank(c E v )/ rank E Question:... + dim Σ + 1 for some appropriate Σ? Seems to work if G line bundle, L E = G s (assume C G v surjective, let Σ = V (image of C G v ))

15 Zach Teitler (BSU) Ranks and generalized ranks 10 / 10 Thank you!