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Symmetric Tensor Decompositions Kristian Ranestad University of Oslo Linz, 26.11.13 Kristian Ranestad Symmetric Tensor Decompositions Given F 2 S d = C [ x 0 , . . . , x n ] d homogeneous of degree d . A presentation F = l d 1 + . . . + l d r

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Symmetric Tensor Decompositions

Kristian Ranestad

University of Oslo

Linz, 26.11.13

Kristian Ranestad Symmetric Tensor Decompositions

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Given F 2 Sd = C[x0, . . . , xn]d homogeneous of degree d. A presentation F = ld

1 + . . . + ld r , with li 2 S1,

is called a Waring decomposition of length r of F.

Kristian Ranestad Symmetric Tensor Decompositions

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Given F 2 Sd = C[x0, . . . , xn]d homogeneous of degree d. A presentation F = ld

1 + . . . + ld r , with li 2 S1,

is called a Waring decomposition of length r of F. Question What is the minimal r, the so called rank r(F) of F, such that F has a Waring decomposition of length r?

Kristian Ranestad Symmetric Tensor Decompositions

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Given F 2 Sd = C[x0, . . . , xn]d homogeneous of degree d. A presentation F = ld

1 + . . . + ld r , with li 2 S1,

is called a Waring decomposition of length r of F. Question What is the minimal r, the so called rank r(F) of F, such that F has a Waring decomposition of length r? How many distinct decompositions of this length does F have?

Kristian Ranestad Symmetric Tensor Decompositions

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Given F 2 Sd = C[x0, . . . , xn]d homogeneous of degree d. A presentation F = ld

1 + . . . + ld r , with li 2 S1,

is called a Waring decomposition of length r of F. Question What is the minimal r, the so called rank r(F) of F, such that F has a Waring decomposition of length r? How many distinct decompositions of this length does F have? Can we find r(F), and if not, why?

Kristian Ranestad Symmetric Tensor Decompositions

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example: conic

A smooth conic in CP2 have equation x2

0 + x2 1 + x2 2 = 0

Kristian Ranestad Symmetric Tensor Decompositions

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example: conic

A smooth conic in CP2 have equation x2

0 + x2 1 + x2 2 = 0

The rank is 3, but in how many ways? x0x1x2 = 0 defines a polar triangle.

Kristian Ranestad Symmetric Tensor Decompositions

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Polar triangle

D E F

Kristian Ranestad Symmetric Tensor Decompositions

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Degenerate polar triangle

D E

Kristian Ranestad Symmetric Tensor Decompositions

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Further degenerate polar triangle

Kristian Ranestad Symmetric Tensor Decompositions

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Apolarity

Sylvester et al. introduced apolarity to find decompositions. Let T = C[y0, . . . , yn] act on S by differentiation: yi(F) = ∂ ∂xi F. Then l = X aixi and g 2 Td ) g(ld) = λg(a0, . . . , an), for some λ 6= 0.

Kristian Ranestad Symmetric Tensor Decompositions

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Apolarity

Sylvester et al. introduced apolarity to find decompositions. Let T = C[y0, . . . , yn] act on S by differentiation: yi(F) = ∂ ∂xi F. Then l = X aixi and g 2 Td ) g(ld) = λg(a0, . . . , an), for some λ 6= 0. Definition g 2 T is apolar to F 2 S if deg g  deg F and g(F) = 0.

Kristian Ranestad Symmetric Tensor Decompositions

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Apolarity

Sylvester et al. introduced apolarity to find decompositions. Let T = C[y0, . . . , yn] act on S by differentiation: yi(F) = ∂ ∂xi F. Then l = X aixi and g 2 Td ) g(ld) = λg(a0, . . . , an), for some λ 6= 0. Definition g 2 T is apolar to F 2 S if deg g  deg F and g(F) = 0. Our key object: F ⊥ = {g 2 T|g(F) = 0} ⇢ T. The quotient T/F ⊥ is Artinian and Gorenstein.

Kristian Ranestad Symmetric Tensor Decompositions

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Apolarity lemma

Let P(S1) and P(T1) denote the projective spaces of 1-dimensional subspaces of S1 (resp. T1). By apolarity, P(S1) = P(T1)∗ and Γ ⇢ P(S1) ) IΓ ⇢ T.

Kristian Ranestad Symmetric Tensor Decompositions

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Apolarity lemma

Let P(S1) and P(T1) denote the projective spaces of 1-dimensional subspaces of S1 (resp. T1). By apolarity, P(S1) = P(T1)∗ and Γ ⇢ P(S1) ) IΓ ⇢ T. Definition Γ ⇢ P(S1) is an apolar subscheme to F if IΓ ⇢ F ⊥. Lemma Let Γ = {[l1], . . . , [lr]} ⇢ P(S1), a collection of r points. Then F = λ1ld

1 + . . . + λrld r

with λi 2 C if and only if IΓ ⇢ F ⊥ ⇢ T.

Kristian Ranestad Symmetric Tensor Decompositions

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example: conic revisited

(x2

0 + x2 1 + x2 2)⊥ = (y0y1, y0y2, y1y2, y 3 0 y 3 1, y 3 1 y 3 2) ⇢ T

Kristian Ranestad Symmetric Tensor Decompositions

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example: conic revisited

(x2

0 + x2 1 + x2 2)⊥ = (y0y1, y0y2, y1y2, y 3 0 y 3 1, y 3 1 y 3 2) ⇢ T

Γ = {[x0], [x1], [x2]} ⇢ P(S1), IΓ = (y0y1, y0y2, y1y2), so Γ is apolar to x2

0 + x2 1 + x2 2.

Kristian Ranestad Symmetric Tensor Decompositions

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Binary forms

F 2 C[x0, x1] ) F ⊥ = (g1, g2) ⇢ C[y0, y1] [Sylvester] where deg g1 + deg g2 = deg F + 2.

Kristian Ranestad Symmetric Tensor Decompositions

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Binary forms

F 2 C[x0, x1] ) F ⊥ = (g1, g2) ⇢ C[y0, y1] [Sylvester] where deg g1 + deg g2 = deg F + 2. F = λ1ld

1 + . . . + λrld r

, I{[l1],...,[lr]} = (g) ⇢ (g1, g2).

Kristian Ranestad Symmetric Tensor Decompositions

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Binary forms

F 2 C[x0, x1] ) F ⊥ = (g1, g2) ⇢ C[y0, y1] [Sylvester] where deg g1 + deg g2 = deg F + 2. F = λ1ld

1 + . . . + λrld r

, I{[l1],...,[lr]} = (g) ⇢ (g1, g2). Assume deg g1  deg g2. r(F) = ⇢ deg g1 when g1 is squarefree deg g2 else

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Cactus rank

Definition The cactus rank (or length) of F is the minimal length of a 0-dimensional apolar subscheme Γ to F, i.e. cr(F) := min{length Γ|dim Γ = 0, IΓ ⇢ F ⊥}.

Kristian Ranestad Symmetric Tensor Decompositions

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Cactus rank

Definition The cactus rank (or length) of F is the minimal length of a 0-dimensional apolar subscheme Γ to F, i.e. cr(F) := min{length Γ|dim Γ = 0, IΓ ⇢ F ⊥}. Clearly cr(F)  r(F) and the inequality may be strict: If d > 2, (x0xd−1

)⊥ = (y2

0 , yd 1 ) ) cr(x0xd−1 1

) = 2 < r(x0xd−1

) = d.

Kristian Ranestad Symmetric Tensor Decompositions

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Border rank

Another much studied rank is the border rank: br(F) = min{r|F is the limit of forms of rank r} The border rank may be smaller than the cactus rank.

Kristian Ranestad Symmetric Tensor Decompositions

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Bounds on the rank

The most famous bound on the rank is not a bound Theorem (Alexander-Hirschowitz 1995) Let F 2 C[x0, . . . , xn] be a general form of degree d, then r(F)= AH(d,n):=d

1 n+1

n+d

e,except

r(F)=n+1 if d=2, r(F)=6,10,15 if d=4, n=2,3,4, r(F)=8 if d=3, n=4. Remark Special forms may have larger rank, a sharp upper bound is only known in a few special cases ((n, d) = (n, 2), (1, d), (2, 3), (2, 4), (3, 3)).

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The simplest lower bound for the rank is explained by

differentiation. If

F = ld

1 + . . . + ld r

and g 2 C[y0, . . . , yn]s then g(F) = λ1ld−s

+ . . . + λrld−s

for some λ1, . . . , λr 2 C

Kristian Ranestad Symmetric Tensor Decompositions

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The simplest lower bound for the rank is explained by

differentiation. If

F = ld

1 + . . . + ld r

and g 2 C[y0, . . . , yn]s then g(F) = λ1ld−s

+ . . . + λrld−s

for some λ1, . . . , λr 2 C So, the dimension h(F, s) of the vector space of partials of order s is a lower bound for r(F). In fact r(F) max{h(F, s)|0 < s < d} The same lower bound is valid for the cactus rank and the border rank.

Kristian Ranestad Symmetric Tensor Decompositions

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An improvement by Landsberg and Teitler depending on the singular locus of the hypersurface V (F). Let d(F, s) be the dimension of the locus of points on V (F) of multiplicity at least s.

Kristian Ranestad Symmetric Tensor Decompositions

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An improvement by Landsberg and Teitler depending on the singular locus of the hypersurface V (F). Let d(F, s) be the dimension of the locus of points on V (F) of multiplicity at least s. Theorem (Landsberg-Teitler 2009) Let F 2 C[x0, . . . , xn]d. Assume that V (F) is not a cone, and let 0 < s < d. Then r(F) h(F, s) + d(F, s) + 1. The proof uses apolarity in a very essential way.

Kristian Ranestad Symmetric Tensor Decompositions

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Bounds on the cactus rank

For n > 2 and d > 6 the cactus rank of a general form is smaller than the rank: Proposition (Bernardi,R 2011) Let F 2 C[x0, . . . , xn]d be any form of degree d, then cr(F)  N(d, n) := ⇢ 2 n+k

when d = 2k + 1

n+k

n+k+1

when d = 2k + 2

Notice that N(d, n) ⇡ O((d/2)n), while AH(d, n) ⇡ O(dn). Question Is this bound sharp? (Yes, at least for cubics (nov 13))

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The proof of the Proposition uses: Theorem (Emsalem 1978) Let Γ ⇢ Cn be a local 0-dimensional scheme with IΓ ⇢ (y1, . . . , yn). Γ is Gorenstein ( ) 9f 2 C[x1, . . . , xn] s.t. IΓ = f ⊥. Furthermore, in this case, length Γ = dimCD(f ) where D(f ) ⇢ C[x1, . . . , xn] is the space of partial derivatives of f

f all orders.

Kristian Ranestad Symmetric Tensor Decompositions

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The proof of the Proposition uses: Theorem (Emsalem 1978) Let Γ ⇢ Cn be a local 0-dimensional scheme with IΓ ⇢ (y1, . . . , yn). Γ is Gorenstein ( ) 9f 2 C[x1, . . . , xn] s.t. IΓ = f ⊥. Furthermore, in this case, length Γ = dimCD(f ) where D(f ) ⇢ C[x1, . . . , xn] is the space of partial derivatives of f

f all orders.

This is relevant for the cactus rank: Lemma (Buczynska-Buczynski, Brachat et al.) If Γ is apolar to F and cr(F) = lengthΓ, then every component of Γ is a local Gorenstein scheme.

Kristian Ranestad Symmetric Tensor Decompositions

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Proof of Proposition

There is a local Gorenstein scheme Γx0(F) for F supported on [x0] 2 P(S1): Let f = F(1, x1, . . . , xn) be the dehomogenization. Then f ⊥ ⇢ C[y1, . . . , yn] defines a local Gorenstein scheme Γx0(F) ⇢ Cn = {y0 6= 0} ⇢ P(S1). It is apolar to F and has length Γx0(F) = dimCD(f ). dimCD(f ) satisfies the bound of the proposition. ⇤

Kristian Ranestad Symmetric Tensor Decompositions

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Proof of Proposition

There is a local Gorenstein scheme Γx0(F) for F supported on [x0] 2 P(S1): Let f = F(1, x1, . . . , xn) be the dehomogenization. Then f ⊥ ⇢ C[y1, . . . , yn] defines a local Gorenstein scheme Γx0(F) ⇢ Cn = {y0 6= 0} ⇢ P(S1). It is apolar to F and has length Γx0(F) = dimCD(f ). dimCD(f ) satisfies the bound of the proposition. ⇤ Remark For any linear form l 2 S1, the homogeneous ideal obtained by saturation of the degree d part of the annihilator (ld+1F)⊥ defines a local Gorenstein scheme Γl of length bounded above by N(d, n) and supported at [l] 2 P(S1).

Kristian Ranestad Symmetric Tensor Decompositions

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Recent improvement

Proposition (Jelisiejew, nov 13) The local Gorenstein scheme Γx0(F) has minimal length among local Gorenstein schemes supported on [x0] 2 P(S1) that are apolar to F.

Kristian Ranestad Symmetric Tensor Decompositions

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Recent improvement

Proposition (Jelisiejew, nov 13) The local Gorenstein scheme Γx0(F) has minimal length among local Gorenstein schemes supported on [x0] 2 P(S1) that are apolar to F. Corollary The cactus rank of F is computed by a decomposition F = F1 + ... + Fr, with linear forms l1, ..., lr, such that cr(F) = length Γl1(F1) + ... + length Γlr (Fr)

Kristian Ranestad Symmetric Tensor Decompositions

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Recent improvement

Proposition (Jelisiejew, nov 13) The local Gorenstein scheme Γx0(F) has minimal length among local Gorenstein schemes supported on [x0] 2 P(S1) that are apolar to F. Corollary The cactus rank of F is computed by a decomposition F = F1 + ... + Fr, with linear forms l1, ..., lr, such that cr(F) = length Γl1(F1) + ... + length Γlr (Fr) Notice, if F = ld, then Γl(F) = [l] 2 P(S1).

Kristian Ranestad Symmetric Tensor Decompositions

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Example

Let F = x2

0x3 + x0x1x2 + x3 1.

Kristian Ranestad Symmetric Tensor Decompositions

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Example

Let F = x2

0x3 + x0x1x2 + x3 1.

Then f = F(1, x1, x2, x3) = x3 + x1x2 + x3

and dimCD(f ) = dimC < x3 + x1x2 + x3

1, x2 + 3x2 1, x1, 1 >= 4.

Kristian Ranestad Symmetric Tensor Decompositions

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Example

Let F = x2

0x3 + x0x1x2 + x3 1.

Then f = F(1, x1, x2, x3) = x3 + x1x2 + x3

and dimCD(f ) = dimC < x3 + x1x2 + x3

1, x2 + 3x2 1, x1, 1 >= 4.

Thus cr(F)  4.

Kristian Ranestad Symmetric Tensor Decompositions

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Example

Let F = x2

0x3 + x0x1x2 + x3 1.

Then f = F(1, x1, x2, x3) = x3 + x1x2 + x3

and dimCD(f ) = dimC < x3 + x1x2 + x3

1, x2 + 3x2 1, x1, 1 >= 4.

Thus cr(F)  4. Furthermore f ⊥ = (y2

1 6y2, y1y2 y3, y2 2 , y2y3, y1y3, y2 3 ),

Kristian Ranestad Symmetric Tensor Decompositions

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Example

Let F = x2

0x3 + x0x1x2 + x3 1.

Then f = F(1, x1, x2, x3) = x3 + x1x2 + x3

and dimCD(f ) = dimC < x3 + x1x2 + x3

1, x2 + 3x2 1, x1, 1 >= 4.

Thus cr(F)  4. Furthermore f ⊥ = (y2

1 6y2, y1y2 y3, y2 2 , y2y3, y1y3, y2 3 ),

so Γx0 ⇠ = Spec C[y1]/(y4

1 ) and also br(F)  4.

Kristian Ranestad Symmetric Tensor Decompositions

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Example

Let F = x2

0x3 + x0x1x2 + x3 1.

Then f = F(1, x1, x2, x3) = x3 + x1x2 + x3

and dimCD(f ) = dimC < x3 + x1x2 + x3

1, x2 + 3x2 1, x1, 1 >= 4.

Thus cr(F)  4. Furthermore f ⊥ = (y2

1 6y2, y1y2 y3, y2 2 , y2y3, y1y3, y2 3 ),

so Γx0 ⇠ = Spec C[y1]/(y4

1 ) and also br(F)  4.

But h(F, 1) = 4 so cr(F), br(F) 4, hence cr(F) = br(F) = 4.

Kristian Ranestad Symmetric Tensor Decompositions

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Example

Let F = x2

0x3 + x0x1x2 + x3 1.

Then f = F(1, x1, x2, x3) = x3 + x1x2 + x3

and dimCD(f ) = dimC < x3 + x1x2 + x3

1, x2 + 3x2 1, x1, 1 >= 4.

Thus cr(F)  4. Furthermore f ⊥ = (y2

1 6y2, y1y2 y3, y2 2 , y2y3, y1y3, y2 3 ),

so Γx0 ⇠ = Spec C[y1]/(y4

1 ) and also br(F)  4.

But h(F, 1) = 4 so cr(F), br(F) 4, hence cr(F) = br(F) = 4. On the other hand, one may show that r(F) = 7.

Kristian Ranestad Symmetric Tensor Decompositions

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Bounds for monomials

Proposition (R-Schreyer 2011) Let 1  d0  d1 . . .  dn, then cr(xd0

0 xd1 1 · · · xdn n ) = (d0 + 1) · · · (dn−1 + 1)

and r(xd0

0 xd1 1 · · · xdn n )  (d1 + 1) · · · (dn + 1).

In particular cr((x0x1 · · · xn)d) = r((x0x1 · · · xn)d) = (d + 1)n. Remark Enrico Carlini, Maria Virginia Catalisano and Anthony V. Geramita proved equality also in the middle.

Kristian Ranestad Symmetric Tensor Decompositions

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In how many ways?

Given r, the set of Waring decompositions {{[l1], . . . , [lr]}|F = ld

1 + . . . + ld r } ⇢ HilbrP(S1)

is a subscheme of the Hilbert scheme. Its closure is called the V(ariety) of S(ums) of P(owers), and denoted VSP(F, r).

Kristian Ranestad Symmetric Tensor Decompositions

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In how many ways?

Given r, the set of Waring decompositions {{[l1], . . . , [lr]}|F = ld

1 + . . . + ld r } ⇢ HilbrP(S1)

is a subscheme of the Hilbert scheme. Its closure is called the V(ariety) of S(ums) of P(owers), and denoted VSP(F, r). For some F and r this variety is a single point, i.e. there is a unique shortest Waring decomposition of F.

Kristian Ranestad Symmetric Tensor Decompositions

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For a general F and r = r(F), dimVSP(F, r) = AH(n, d)(n + 1) ✓n + d n ◆ .

((n, d) 6= (n, 2), (4, 3), (2, 4), (3, 4), (4, 4)) Kristian Ranestad Symmetric Tensor Decompositions

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For a general F and r = r(F), dimVSP(F, r) = AH(n, d)(n + 1) ✓n + d n ◆ .

((n, d) 6= (n, 2), (4, 3), (2, 4), (3, 4), (4, 4))

Theorem (Sylvester, Hilbert, Palatini, Richmond, 1851-1902, Mukai, Dolgachev-Kanev, Schreyer, Iliev, R 1989-2000) If F is general and r = r(F), then VSP(F, r) is a point if (n, d) = (1, 2r 1), (2, 5), (3, 3), P1 if (n, d) = (1, 2r 2), P2 when (n, d) = (2, 3), a K3 surface when (n, d) = (2, 6), a Fano threefold when (n, d) = (2, 4), a Fano fivefold when (n, d) = (4, 3), a Hyperk¨ ahler fourfold when (n, d) = (5, 3).

Kristian Ranestad Symmetric Tensor Decompositions

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algorithms/obstructions

Some methods have been developed, building on apolarity, to find Waring decompositions of F. (cf. [Brachat et al.] and [Oeding-Ottaviani]) For small n and d or when r(F) << AH(n, d), then these methods are effective. In computations, one normally works over Q or a finite field. For general F, the first obstruction is therefore to find a point on the variety VSP(F, r) with the additional property that each l is defined over the ground field. Question VSP(x3

0x1 + x3 1x2 + x3 2x0, 6) is Fano threefold. Does there exist

l1, . . . , l6 2 Q[x0, x1, x2] and rational numbers λ1, . . . , λ6 such that x3

0x1 + x3 1x2 + x3 2x0 = λ1l4 1 + . . . + λ6l4 6?

Kristian Ranestad Symmetric Tensor Decompositions

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VSP and VAPS

VSP(F, r) is a natural subscheme of VAPS(F, r) = {Γ ⇢ P(S1))|IΓ ⇢ F ⊥} ⇢ HilbrP(S1). VSP(F, r) ⇢ VAPS(F, r) is the closure of the set of smooth apolar subschemes.

Kristian Ranestad Symmetric Tensor Decompositions

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VSP and VAPS

VSP(F, r) is a natural subscheme of VAPS(F, r) = {Γ ⇢ P(S1))|IΓ ⇢ F ⊥} ⇢ HilbrP(S1). VSP(F, r) ⇢ VAPS(F, r) is the closure of the set of smooth apolar subschemes. In general VSP(F, r) is a proper subscheme of VAPS(F, r), in particular when r(F) > N(n, d) cr(F).

Kristian Ranestad Symmetric Tensor Decompositions

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VSP and VAPS

VSP(F, r) is a natural subscheme of VAPS(F, r) = {Γ ⇢ P(S1))|IΓ ⇢ F ⊥} ⇢ HilbrP(S1). VSP(F, r) ⇢ VAPS(F, r) is the closure of the set of smooth apolar subschemes. In general VSP(F, r) is a proper subscheme of VAPS(F, r), in particular when r(F) > N(n, d) cr(F). The difference VASP(F, r) \ VSP(F, r) is a second obstruction to finding a decomposition of F.

Kristian Ranestad Symmetric Tensor Decompositions

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VSP and VAPS

VSP(F, r) is a natural subscheme of VAPS(F, r) = {Γ ⇢ P(S1))|IΓ ⇢ F ⊥} ⇢ HilbrP(S1). VSP(F, r) ⇢ VAPS(F, r) is the closure of the set of smooth apolar subschemes. In general VSP(F, r) is a proper subscheme of VAPS(F, r), in particular when r(F) > N(n, d) cr(F). The difference VASP(F, r) \ VSP(F, r) is a second obstruction to finding a decomposition of F. Even when F is a quadric of rank n, VAPS(F, n) \ VSP(F, n) 6= ; when n >> 0

Kristian Ranestad Symmetric Tensor Decompositions

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Quadrics

Let Q 2 C[x0, . . . , xn] be a quadric of maximal rank n + 1. If Q = l2

0 + . . . + l2 n, then {l0 · · · ln = 0} ⇢ P(T1)

is isomorphic to the standard coordinate simplex. It is classically known as a polar simplex.

Kristian Ranestad Symmetric Tensor Decompositions

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Quadrics

Let Q 2 C[x0, . . . , xn] be a quadric of maximal rank n + 1. If Q = l2

0 + . . . + l2 n, then {l0 · · · ln = 0} ⇢ P(T1)

is isomorphic to the standard coordinate simplex. It is classically known as a polar simplex. Theorem (R-Schreyer 2011) Let Q 2 C[x0, . . . , xn] be a quadric of rank n + 1. VSP(Q, n + 1) is rational variety of dimension n+1

It is a smooth Fano variety of index 2 with Picard group isomorphic to Z if n < 5. VSP(Q, n + 1) is singular if n 5. When n 23, VAPS(Q, n + 1) is reducible.

Kristian Ranestad Symmetric Tensor Decompositions

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Thank You!

Kristian Ranestad Symmetric Tensor Decompositions

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References

Alessandra Bernardi, KR: The rank of cubic forms arXiv:1110.2197

J. Brachat, P. Comon, B. Mourrain, E. Tsigaridas:Symmetric Tensor decompositions, arXiv:0901.3706

Jaroslaw Buczynski, Adam Ginensky, J.M. Landsberg: Determinental equations for secant varieties and the Eisenbud-Koh-Stillman conjecture, arXiv:1007.0192 Weronika Buczynska, Jaroslaw Buczynski: Secant varieties to high degree Veronese reembeddings, catalecticant matrices and smoothable Gorenstein schemes, arXiv:1012.3563 Enrico Carlini, Maria Virginia Catalisano, Anthony V. Geramita: The Solution to Waring’s Problem for Monomials, arXiv:1110.0745 Tony Iarrobino and Vassil Kanev: Power Sums, Gorenstein Algebras, and Determinantal Loci, Springer Lecture Notes in Mathematics 1721, Springer (1999) Joseph Landsberg and Giorgio Ottaviani:Equations for secant varieties to Veronese varieties, arXiv:1010.1825 Luke Oeding and Giorgio Ottaviani:Eigenvectors of Tensors and algorithms for Waring Decomposition, arXiv:1103.0203 Claudiu Raicu: 3 ⇥ 3 Minors of Catalecticants, arXiv:1011.1564 KR, Frank Olaf Schreyer: On the rank of a symmetric form, arXiv:1104.3648 KR, Frank Olaf Schreyer: The Variety of Polar Simplices, arXiv:1104.2728 Kristian Ranestad Symmetric Tensor Decompositions