Direct Sum Decomposability of Polynomials

Transcription

1 Direct Sum Decomposability of Polynomials Zach Teitler Interactions between Commutative Algebra and Algebraic Geometry II September 28, 2013 Joint work with Weronika Buczyńska, Jarek Buczyński, Johannes Kleppe 1 / 21

2 Outline 1 Apolarity 2 Problem 1: Direct sum decomposability 3 Problem 2: Bounds for Waring rank 4 Results 5 Flatness 6 Proofs 7 Low-degree generators 2 / 21

3 Apolarity S = C[x 1,..., x n ], T = C[ 1,..., n ] dual ring: i acts as / x i. For F S, F T ideal. Example x a 1 1 x an n = a 1+1 1,..., an+1 n. F defining equation of mirror arrangement of finite complex reflection group G = F generated by invariants of G [Steinberg, Shephard Todd, Chevalley] 3 / 21

4 Apolarity of Determinants Example (Shafiei, 2012) x 1,1... x 1,n det n = det.. x n,1... x n,n det n is generated by: i,j 2, i,j 1 i,j2, i1,j i2,j (two entries from same row or column) permanents ( of ) 2 2 submatrices: i,j per i,l = i,j k,l + i,l k,j. k,j k,l See [Shafiei, 2012], [Shafiei, 2013] for more: permanents, Pfaffians, symmetric determinants, etc. 4 / 21

5 Direct sum decompositions Can F = F (x 1,..., x n ) be written as a sum of polynomials in separate variables? We allow a linear change of coordinates: F = G(t 1,..., t k ) + H(t k+1,..., t n ) where the t i are linearly independent linear forms. Example xy G(x) + H(y), but xy = 1 4 (x + y)2 1 4 (x y)2. Let det n = det((x ij ) 1 i,j n ), the n n generic determinant. Problem det 2 = ad bc is decomposable. Is det n decomposable for n > 2? 5 / 21

6 Topology and Direct Sums A, B complex manifolds of the same dimension cohomology rings H (A) = C[ X ]/F, H (Y ) = C[ Y ]/G connected sum A#B H (A#B) = C[ X, Y ]/(F (X ) + G(Y )). If H (M) = C[ T ]/P and P is indecomposable as a direct sum, then M is indecomposable as a connected sum. 6 / 21

7 Waring rank Waring rank: r(f ) = least r such that F = l d ld r. Quadratic forms: Waring rank equals rank of matrix. Binary forms: [Sylvester, 1851], [Gundelfinger, 1886],.... x 1 x n = 1 2 n 1 n! ±(x1 ± x 2 ± ± x n ) n so r(x 1 x n ) 2 n 1. Turns out to be equal. [Ranestad-Schreyer, 2011] r(x a 1 1 x n an ) = (1 + a 1 ) (1 + a n 1 ) (a 1 a n ) [Carlini-Catalisano-Geramita, 2011] det n = sum of n! terms of the form x 1 x n. So r(det n ) 2 n 1 n!. 7 / 21

8 Lower bounds for rank Sylvester: r(f ) max{dim(t /F ) a } Landsberg-T.: A bound in terms of singularities of V (F ). Ranestad-Schreyer: r(f ) 1 δ a dim(t /F ) a, where δ = maximum degree of a generator of F. 8 / 21

9 Example F = x 1 x n has dim(t /F ) a = ( n a), (x1 x n ) = 2 1,..., 2 n. x 1 x n = 1 2 n 1 ±(x1 ± x n! 2 ± x n ) n = r(x 1 x n ) 2 n 1 Sylvester: r(x 1 x n ) ( n n/2) 2 n 1 / n (r(xyz) 3) Ranestad-Schreyer: r(x 1 x n ) 1 2 ( n a) = 2 n 1. 9 / 21

10 Example dim(t / det n ) a = ( n a) 2, (detn ) generated by quadrics. r(det n ) 2 n 1 n! (r(det 3 ) 24) Sylvester: r(det n ) ( n n/2 ) 2 (r(det 3 ) 9) Ranestad-Schreyer-Shafiei: r(det n ) 1 2( 2n n ) (r(det 3 ) 10) (Landsberg-T.: r(det 3 ) 14, but Ranestad-Schreyer-Shafiei grows more quickly.) 10 / 21

11 Apolar generating degree What is the generating degree of F? Say deg F = d. Then F contains all differential operators of degree > d, so no generators of degree d + 2 or higher (but maybe d + 1). Problem Give conditions for F to have high-degree or low-degree generators. Theorem (Casnati Notari) F has a minimal generator of degree d + 1 if and only if r(f ) = 1, F = x d. 11 / 21

12 Theorem (Buczyńska-Buczyński-Kleppe-T.) If F is decomposable as a direct sum then F has a minimal generator of degree d. Corollary For n > 2, x 1 x n and det n are not decomposable. Proof. Say F = G(X ) H(Y ) is a direct sum decomposition. For 0 a < d, F a = G a H a : DG = DH C[X ] C[Y ] = either DG = DH = 0 or deg D = d. Let X (G) = 1, Y (H) = 1, = X + Y. Then Fd = (G H ) d +. So F = (G H ) + and is a minimal generator. 12 / 21

13 Converse The converse fails. Example F = xy d 1 has F = α 2, β d. But F is indecomposable because l d 1 ld 2 However xy d 1 is a limit of direct sums: has distinct factors. xy d 1 1 ( = lim (y + tx) d y d). t 0 dt Theorem (BBKT) If F has a minimal generator of degree d then F is a limit of direct sums. 13 / 21

14 Converse again Theorem (BBKT) If F has a minimal generator of degree d then F is a limit of direct sums. Once again the converse fails! Example x d ty d x d, but (x d ) = α d+1, β. xyz tw 3 xyz, but (xyz) = α 2, β 2, γ 2, δ. Theorem (BBKT) Let n 2, d 3. If F is a limit of direct sums and F cannot be written using fewer variables, then F has a minimal generator of degree d. 14 / 21

15 Chart of inclusions DirSum ApoMax DirSum DirSum Con ApoMax Con = DirSum Con DirSum: decomposable as a direct sum ApoMax: F has a generator of maximal degree d Con: concise, i.e., cannot be written using fewer variables 15 / 21

16 An idea that doesn t work Suppose F t F 0 = F and the F t are direct sums for t 0. Does semicontinuity of graded Betti numbers show that F has a generator of degree d? No, because F is not necessarily the flat limit of the F t. Example (x d ) = α d+1, β (x d ty d ) = αβ, tα d + β d αβ, tα d + β d αβ, α d+1, β d So (x d ty d ) (x d ). 16 / 21

17 Well, it works sometimes Theorem (BBKT) For d = 3, if F t F and F is concise then F t F is always flat. But For n = 3, if F has a minimal generator of degree d then there exists some family F t F, F t direct sums, such that Ft F is flat. There exists F such that F has a minimal generator of degree d, so F is a limit of direct sums; but for every family of direct sums F t F, Ft F is not flat. The last item is a consequence of the existence of non-smoothable Gorenstein schemes. This forces trickier proofs for the previous theorems. 17 / 21

18 An idea that does work Theorem (BBKT) If F has a minimal generator of degree d then F is a limit of direct sums. Proof. Suppose F has a minimal generator of degree d. By Gorenstein duality, F has a high degree syzygy. By Koszul homology, F contains quadratic generators: the 2 2 minors of a 2 n matrix L of linear forms. L L, endomorphism of space of linear forms Jordan normal form of L either gives a direct sum decomposition of F, or (if L is nilpotent) a limit of direct sums. 18 / 21

19 Theorem (BBKT) Let n 2, d 3. If F is a limit of direct sums and F cannot be written using fewer variables, then F has a minimal generator of degree d. Proof. Suppose F is a concise limit of direct sums, F t F. Let J = lim F t. J F. Each F t has a minimal generator of degree d By Gorenstein duality β n 1,n (F t ) > 0 By semicontinuity of graded Betti numbers, β n 1,n (J) > 0 This syzygy lies in the minimal-degree strand by conciseness (no linear generators). So F has the same high degree syzygy Hence F also has a minimal generator of degree d. 19 / 21

20 Low-degree generators Theorem (BBKT) If F is a homogeneous form of degree d in n variables and δ is the generating degree of F then d (δ 1)n. If F is generated by quadrics then d n. Question What are the forms F such that d = n and F is generated by quadrics? F = x 1 x n has F = α 2 1,..., α2 n. 20 / 21

21 Thank you! 21 / 21