Ranks of polynomials
Zach Teitler
March 27, 2014
1 / 25
Sums of powers
In 1770, Edward Waring claimed every natural number is a sum of at most
4 squares, at most 9 cubes, at most 19 fourth powers, “and so on.”
2014 = 422 + 152 + 52
2014 = 93 + 83 + 63 + 63 + 63 + 53
2 / 25
Sums of powers
In 1770, Edward Waring claimed every natural number is a sum of at most
4 squares, at most 9 cubes, at most 19 fourth powers, “and so on.”
2014 = 422 + 152 + 52
2014 = 93 + 83 + 63 + 63 + 63 + 53
2 / 25
Polynomials as sums of powers
F (x1 , . . . , xn ): polynomial of degree d in n variables
Goal:
F (x1 , . . . , xn ) = c1 `d1 + · · · + cr `dr
where `i are linear.
3 / 25
Example: power sum decompositions of xy
4 / 25
Example: power sum decompositions of xy
xy =
1
(x + y )2 − x 2 − y 2
2
using three squares
y
x
x
y
4 / 25
Example: power sum decompositions of xy
xy =
1
(x + y )2 − x 2 − y 2
2
xy =
using two squares
using three squares
y
1
(x + y )2 − (x − y )2
4
x
x −y
x
y
x
y
x
y
4 / 25
Waring rank
xy =
xyz =
1
(x + y )2 − (x − y )2
4
1
(x + y + z)3 − (x + y − z)3 − (x + z − y )3 − (y + z − x)3
24
Picture???
Waring rank: r (F ) = least number of terms
in a power sum decomposition F = c1 `d1 + · · · + cr `dr
5 / 25
Waring rank
xy =
xyz =
1
(x + y )2 − (x − y )2
4
1
(x + y + z)3 − (x + y − z)3 − (x + z − y )3 − (y + z − x)3
24
Picture???
Waring rank: r (F ) = least number of terms
in a power sum decomposition F = c1 `d1 + · · · + cr `dr
5 / 25
Waring rank
xy =
xyz =
1
(x + y )2 − (x − y )2
4
1
(x + y + z)3 − (x + y − z)3 − (x + z − y )3 − (y + z − x)3
24
Picture???
Waring rank: r (F ) = least number of terms
in a power sum decomposition F = c1 `d1 + · · · + cr `dr
5 / 25
Motivation
Waring rank: r (F ) = least number of terms
in a power sum decomposition F = c1 `d1 + · · · + cr `dr
• Generalization of SVD
d = 2: For quadratic forms,
Waring rank = matrix rank.
• Measure of complexity
• Signal processing
• Mixture models in statistics
6 / 25
Motivation
Waring rank: r (F ) = least number of terms
in a power sum decomposition F = c1 `d1 + · · · + cr `dr
• Generalization of SVD
• Measure of complexity
`d “simple”
r (F ) = “complexity”
• Signal processing
• Mixture models in statistics
6 / 25
Motivation
Waring rank: r (F ) = least number of terms
in a power sum decomposition F = c1 `d1 + · · · + cr `dr
• Generalization of SVD
• Measure of complexity
• Signal processing
`d ! “single source”
Blind Source Separation
• Mixture models in statistics
6 / 25
Motivation
Waring rank: r (F ) = least number of terms
in a power sum decomposition F = c1 `d1 + · · · + cr `dr
• Generalization of SVD
• Measure of complexity
• Signal processing
• Mixture models in statistics
`d ! joint distribution
Pof dd iid random variables
ci `i ! mixture model.
6 / 25
Example: Quadratic form
5
5
x 2 + 5xy + 4y 2 = (x)2 + (x + y )2 − (x − y )2 + 4(y )2
4
4
1
1
2
= (x + 2y ) + (x + y )2 − (x − y )2 .
4
4
But
2
2
x + 5xy + 4y = x
y
1 5/2
x
.
5/2 4
y
Diagonalize:
1 5/2
0
t σ1
=B
B
5/2 4
0 σ2
so
x 2 + 5xy + 4y 2 = σ1 (b11 x + b12 y )2 + σ2 (b21 x + b22 y )2 .
Explicitly:
9
1
x 2 + 5xy + 4y 2 = (x + 2y )2 − (x − 2y )2 .
8
8
7 / 25
Example: Quadratic form
5
5
x 2 + 5xy + 4y 2 = (x)2 + (x + y )2 − (x − y )2 + 4(y )2
4
4
1
1
2
= (x + 2y ) + (x + y )2 − (x − y )2 .
4
4
But
2
2
x + 5xy + 4y = x
y
1 5/2
x
.
5/2 4
y
Diagonalize:
1 5/2
0
t σ1
=B
B
5/2 4
0 σ2
so
x 2 + 5xy + 4y 2 = σ1 (b11 x + b12 y )2 + σ2 (b21 x + b22 y )2 .
Explicitly:
9
1
x 2 + 5xy + 4y 2 = (x + 2y )2 − (x − 2y )2 .
8
8
7 / 25
Example: Quadratic form
5
5
x 2 + 5xy + 4y 2 = (x)2 + (x + y )2 − (x − y )2 + 4(y )2
4
4
1
1
2
= (x + 2y ) + (x + y )2 − (x − y )2 .
4
4
But
2
2
x + 5xy + 4y = x
y
1 5/2
x
.
5/2 4
y
Diagonalize:
1 5/2
0
t σ1
=B
B
5/2 4
0 σ2
so
x 2 + 5xy + 4y 2 = σ1 (b11 x + b12 y )2 + σ2 (b21 x + b22 y )2 .
Explicitly:
9
1
x 2 + 5xy + 4y 2 = (x + 2y )2 − (x − 2y )2 .
8
8
7 / 25
Example: Quadratic form
5
5
x 2 + 5xy + 4y 2 = (x)2 + (x + y )2 − (x − y )2 + 4(y )2
4
4
1
1
2
= (x + 2y ) + (x + y )2 − (x − y )2 .
4
4
But
2
2
x + 5xy + 4y = x
y
1 5/2
x
.
5/2 4
y
Diagonalize:
1 5/2
0
t σ1
=B
B
5/2 4
0 σ2
so
x 2 + 5xy + 4y 2 = σ1 (b11 x + b12 y )2 + σ2 (b21 x + b22 y )2 .
Explicitly:
9
1
x 2 + 5xy + 4y 2 = (x + 2y )2 − (x − 2y )2 .
8
8
7 / 25
Example: Dice
1
1
D6 = x1 + · · · + x6
6
6
D20 =
1
1
x1 + · · · + x20
20
20
P({i, j}) = coefficient of xi xj in D6 D20
Question: Can we get the same probability distribution by rolling the
same die twice?
D∗ = p1 x1 + · · · + p20 x20
No, D∗2 = D6 D20 is impossible.
8 / 25
Example: Dice
1
1
D6 = x1 + · · · + x6
6
6
D20 =
1
1
x1 + · · · + x20
20
20
P({i, j}) = coefficient of xi xj in D6 D20
Question: Can we get the same probability distribution by rolling the
same die twice?
D∗ = p1 x1 + · · · + p20 x20
No, D∗2 = D6 D20 is impossible.
8 / 25
Example: Dice
1
1
D6 = x1 + · · · + x6
6
6
D20 =
1
1
x1 + · · · + x20
20
20
P({i, j}) = coefficient of xi xj in D6 D20
Question: Can we get the same probability distribution by rolling the
same die twice?
D∗ = p1 x1 + · · · + p20 x20
No, D∗2 = D6 D20 is impossible.
8 / 25
Example: Dice, continued
q[
[ x
D[ = p1[ x1 + · · · + p20
20
q]
]
D] = p1] x1 + · · · + p20
x20
D6 D20 = q[ D[2 + q] D]2 ?
9 / 25
Questions
Given F ,
• What is r (F )?
• Find `i such that F =
P
ci `di .
• Describe the set of solutions, {(`1 , . . . , `r ) | F =
P
ci `di }.
10 / 25
Quadratic and binary forms
Waring ranks are completely known for degree 2 polynomials, and for
polynomials in 2 variables:
If F is any quadratic form (degree 2 polynomial)
then F can be represented by a symmetric matrix,
and the Waring rank of F is equal to the rank of that matrix.
If F is a polynomial in 2 variables then the Waring rank of F is known,
thanks to work by J.J. Sylvester in 1851, and others.
Other than that: r (F ) is only known for a handful of polynomials F !
11 / 25
Example: xyz
xyz =
1
(x + y + z)3 − (x + y − z)3 − (x + z − y )3 − (y + z − x)3
24
so
r (xyz) ≤ 4
Since xyz is not a perfect square,
r (xyz) ≥ 2
12 / 25
Catalecticant
F ∈ C[x1 , . . . , xn ]d = polynomials of degree d in xi ’s
D ∈ C[∂1 , . . . , ∂n ]a = differential operators of order a
(constant coefficients)
For a fixed F , 0 ≤ a ≤ d, and variable D:
C[∂1 , . . . , ∂n ]a → C[x1 , . . . , xn ]d−a
D 7→ DF
Called the a’th catalecticant of F , CFa , a linear map
Theorem (Sylvester, 1851)
r (F ) ≥ rank CFa
13 / 25
Catalecticant
F ∈ C[x1 , . . . , xn ]d = polynomials of degree d in xi ’s
D ∈ C[∂1 , . . . , ∂n ]a = differential operators of order a
(constant coefficients)
For a fixed F , 0 ≤ a ≤ d, and variable D:
C[∂1 , . . . , ∂n ]a → C[x1 , . . . , xn ]d−a
D 7→ DF
Called the a’th catalecticant of F , CFa , a linear map
Theorem (Sylvester, 1851)
r (F ) ≥ rank CFa
13 / 25
Catalecticant
F ∈ C[x1 , . . . , xn ]d = polynomials of degree d in xi ’s
D ∈ C[∂1 , . . . , ∂n ]a = differential operators of order a
(constant coefficients)
For a fixed F , 0 ≤ a ≤ d, and variable D:
C[∂1 , . . . , ∂n ]a → C[x1 , . . . , xn ]d−a
D 7→ DF
Called the a’th catalecticant of F , CFa , a linear map
Theorem (Sylvester, 1851)
r (F ) ≥ rank CFa
13 / 25
Catalecticant
F ∈ C[x1 , . . . , xn ]d = polynomials of degree d in xi ’s
D ∈ C[∂1 , . . . , ∂n ]a = differential operators of order a
(constant coefficients)
For a fixed F , 0 ≤ a ≤ d, and variable D:
C[∂1 , . . . , ∂n ]a → C[x1 , . . . , xn ]d−a
D 7→ DF
Called the a’th catalecticant of F , CFa , a linear map
Theorem (Sylvester, 1851)
r (F ) ≥ rank CFa
13 / 25
Example: Catalecticant of xyz
So far:
∂y ∂z xyz = x
2 ≤ r (xyz) ≤ 4
∂x ∂z xyz = y
∂x ∂y xyz = z
2
) = hx, y , zi
image(Cxyz
2
r (xyz) ≥ rank(Cxyz
)=3
14 / 25
Example: Catalecticant of xyz
So far:
∂y ∂z xyz = x
2 ≤ r (xyz) ≤ 4
∂x ∂z xyz = y
∂x ∂y xyz = z
2
) = hx, y , zi
image(Cxyz
2
r (xyz) ≥ rank(Cxyz
)=3
14 / 25
Example: Catalecticant of xyz
So far:
∂y ∂z xyz = x
2 ≤ r (xyz) ≤ 4
∂x ∂z xyz = y
∂x ∂y xyz = z
2
) = hx, y , zi
image(Cxyz
2
r (xyz) ≥ rank(Cxyz
)=3
14 / 25
Example: Catalecticant of xyz
So far:
∂y ∂z xyz = x
2 ≤ r (xyz) ≤ 4
∂x ∂z xyz = y
∂x ∂y xyz = z
2
) = hx, y , zi
image(Cxyz
2
r (xyz) ≥ rank(Cxyz
)=3
14 / 25
Interlude: That crazy name
J.J. Sylvester wrote poetry in English, Latin, and Italian, and he enjoyed
prosody, the technical study of poetry.
• A line of poetry is “catalectic” if it is shorter than the poem’s form
• When the catalecticant of F has lower rank, then the power sum
decomposition of F is shorter
Meicatalecticizant would more completely express the meaning of
that which, for the sake of brevity, I denominate the
catalecticant.
— J.J. Sylvester, 1852
To his credit, in the same paper, Sylvester introduced the term
“unimodular” in its current meaning.
— B. Reznick
15 / 25
Interlude: That crazy name
J.J. Sylvester wrote poetry in English, Latin, and Italian, and he enjoyed
prosody, the technical study of poetry.
• A line of poetry is “catalectic” if it is shorter than the poem’s form
• When the catalecticant of F has lower rank, then the power sum
decomposition of F is shorter
Meicatalecticizant would more completely express the meaning of
that which, for the sake of brevity, I denominate the
catalecticant.
— J.J. Sylvester, 1852
To his credit, in the same paper, Sylvester introduced the term
“unimodular” in its current meaning.
— B. Reznick
15 / 25
Interlude: That crazy name
J.J. Sylvester wrote poetry in English, Latin, and Italian, and he enjoyed
prosody, the technical study of poetry.
• A line of poetry is “catalectic” if it is shorter than the poem’s form
• When the catalecticant of F has lower rank, then the power sum
decomposition of F is shorter
Meicatalecticizant would more completely express the meaning of
that which, for the sake of brevity, I denominate the
catalecticant.
— J.J. Sylvester, 1852
To his credit, in the same paper, Sylvester introduced the term
“unimodular” in its current meaning.
— B. Reznick
15 / 25
Improved lower bound
An improved lower bound
• In terms of singularities
• Hypothesis: F can’t be written using fewer variables
16 / 25
Singularities
Σa (F ) = { points where F vanishes with multiplicity > a }
i.e., where F , first derivatives, second derivatives, . . . , a’th derivatives
vanish.
Σ1 (F ) = {singularities} = {non-manifold points of F −1 (0)}
17 / 25
Conciseness
F called concise if can’t be written using fewer variables
• Example: F (x, y , z) = (x + y )z is not concise:
rewrite as F (a, b, z) = F (a, z) = az where a = x + y , b = x − y
Equivalent:
• F is concise
• Σd−1 (F ) = {origin}
• CFd−1 surjective
18 / 25
Conciseness
F called concise if can’t be written using fewer variables
• Example: F (x, y , z) = (x + y )z is not concise:
rewrite as F (a, b, z) = F (a, z) = az where a = x + y , b = x − y
Equivalent:
• F is concise
• Σd−1 (F ) = {origin}
• CFd−1 surjective
18 / 25
Improved lower bound
Theorem (Landsberg–T. ’10)
If F is concise then
r (F ) ≥ rank CFa + dim Σd−a .
19 / 25
Example: xyz
xyz = 0: union of coordinate planes
• Non-manifold points: coordinate axes
• dimension = 1
So
2
r (xyz) ≥ rank Cxyz
+ dim Σ1 = 3 + 1 = 4.
Therefore r (xyz) = 4.
20 / 25
Idea of proof
Suppose F = `d1 + · · · + `dr .
• Let L = forms of degree d − a that vanish at each `i
• r equations, so codim L ≤ r
• L ⊆ ker CFd−a
• r ≥ codim L ≥ codim ker CFd−a = rank CFd−a
This proves Sylvester’s theorem.
Next show that L is disjoint from a certain “bad set”
so its codimension must be even greater.
21 / 25
Idea of proof
Suppose F = `d1 + · · · + `dr .
• Let L = forms of degree d − a that vanish at each `i
• r equations, so codim L ≤ r
• L ⊆ ker CFd−a
• r ≥ codim L ≥ codim ker CFd−a = rank CFd−a
This proves Sylvester’s theorem.
Next show that L is disjoint from a certain “bad set”
so its codimension must be even greater.
21 / 25
Idea of proof
Suppose F = `d1 + · · · + `dr .
• Let L = forms of degree d − a that vanish at each `i
• r equations, so codim L ≤ r
• L ⊆ ker CFd−a
• r ≥ codim L ≥ codim ker CFd−a = rank CFd−a
This proves Sylvester’s theorem.
Next show that L is disjoint from a certain “bad set”
so its codimension must be even greater.
21 / 25
Idea of proof
Suppose F = `d1 + · · · + `dr .
• Let L = forms of degree d − a that vanish at each `i
• r equations, so codim L ≤ r
• L ⊆ ker CFd−a
• r ≥ codim L ≥ codim ker CFd−a = rank CFd−a
This proves Sylvester’s theorem.
Next show that L is disjoint from a certain “bad set”
so its codimension must be even greater.
21 / 25
Idea of proof
Suppose F = `d1 + · · · + `dr .
• Let L = forms of degree d − a that vanish at each `i
• r equations, so codim L ≤ r
• L ⊆ ker CFd−a
• r ≥ codim L ≥ codim ker CFd−a = rank CFd−a
This proves Sylvester’s theorem.
Next show that L is disjoint from a certain “bad set”
so its codimension must be even greater.
21 / 25
Idea of proof
Suppose F = `d1 + · · · + `dr .
• Let L = forms of degree d − a that vanish at each `i
• r equations, so codim L ≤ r
• L ⊆ ker CFd−a
• r ≥ codim L ≥ codim ker CFd−a = rank CFd−a
This proves Sylvester’s theorem.
Next show that L is disjoint from a certain “bad set”
so its codimension must be even greater.
21 / 25
Idea of proof
Suppose F = `d1 + · · · + `dr .
• Let L = forms of degree d − a that vanish at each `i
• r equations, so codim L ≤ r
• L ⊆ ker CFd−a
• r ≥ codim L ≥ codim ker CFd−a = rank CFd−a
This proves Sylvester’s theorem.
Next show that L is disjoint from a certain “bad set”
so its codimension must be even greater.
21 / 25
Example: Products of variables
Let F = x1 · · · xn =
• r (F ) ≤
1
n!2n−1
X
(x1 + 2 x2 + · · · + n xn )n 2 · · · n
i =±1
2n−1
• Image of CFa spanned by products of subsets of the variables
• rank CFa = na
• dim Σa = n − a − 1
E.g.:
3 ≤ 4 ≤ r (xyz) ≤ 4
6 ≤ 7 ≤ r (xyzw ) ≤ 8
Turns out r (x1 · · · xn ) = 2n−1 (Ranestad–Schreyer, ’12)
22 / 25
Example: Products of variables
Let F = x1 · · · xn =
• r (F ) ≤
1
n!2n−1
X
(x1 + 2 x2 + · · · + n xn )n 2 · · · n
i =±1
2n−1
• Image of CFa spanned by products of subsets of the variables
• rank CFa = na
• dim Σa = n − a − 1
E.g.:
3 ≤ 4 ≤ r (xyz) ≤ 4
6 ≤ 7 ≤ r (xyzw ) ≤ 8
Turns out r (x1 · · · xn ) = 2n−1 (Ranestad–Schreyer, ’12)
22 / 25
Example: Products of variables
Let F = x1 · · · xn =
• r (F ) ≤
1
n!2n−1
X
(x1 + 2 x2 + · · · + n xn )n 2 · · · n
i =±1
2n−1
• Image of CFa spanned by products of subsets of the variables
• rank CFa = na
• dim Σa = n − a − 1
E.g.:
3 ≤ 4 ≤ r (xyz) ≤ 4
6 ≤ 7 ≤ r (xyzw ) ≤ 8
Turns out r (x1 · · · xn ) = 2n−1 (Ranestad–Schreyer, ’12)
22 / 25
Example: Determinant
Let detn = determinant of n × n matrix with variables xi,j
• r (detn ) ≤ n!2n−1
a
• Image of Cdet
spanned by minors
n
2
a
• rank Cdet
= na
n
• Σa = matrices of rank < n − a, dimension = n2 − (a + 1)2
E.g.:
9 ≤ 14 ≤ r (det3 ) ≤ 24
That’s almost all we know about ranks of determinants!
23 / 25
Waring rank is finite
How do we know Waring rank is finite?
Fix a monomial x1a1 · · · xnan of degree d = a1 + · · · + an :
1
d!2d−1
X
1 · · · d−1 (1 y1 + · · · + d−1 yd−1 + yd )d
1 ,...,d−1 =±1
= y1 · · · yd
Since every monomial can be written as a finite combination of powers, so
can every polynomial.
24 / 25
Waring rank is finite
How do we know Waring rank is finite?
Fix a monomial x1a1 · · · xnan of degree d = a1 + · · · + an :
1
d!2d−1
X
1 · · · d−1 (1 y1 + · · · + d−1 yd−1 + yd )d
1 ,...,d−1 =±1
= y1 · · · ya1 ya1 +1 · · · ya1 +a2 · · · · · · yd
| {z }
{z
}
| {z } |
x1
x2
xn
Since every monomial can be written as a finite combination of powers, so
can every polynomial.
24 / 25
Upper bounds for Waring rank
The vector space of polynomials has dimension
n+d−1
n−1
.
Upper bound for rank
n+d−1
n−1
n+d−1
−n+1
n−1
n+d−2
n−1
n+d−2
− n+d−6
n−1
n−3
n+d−2
− n+d−6
n−1
n−3
2d n1 n+d−1
n−1 e
Basis of powers
(Geramita ’96; Landsberg–T. ’10)
(Bialynicki-Birula–Schinzel ’08)
(Jelisiejew ’13)
−
n+d−7
n−3
(Ballico–De Paris ’13)
(Blekherman–T. ’14)
None of these is sharp.
25 / 25
Upper bounds for Waring rank
The vector space of polynomials has dimension
n+d−1
n−1
.
Upper bound for rank
n+d−1
n−1
n+d−1
−n+1
n−1
n+d−2
n−1
n+d−2
− n+d−6
n−1
n−3
n+d−2
− n+d−6
n−1
n−3
2d n1 n+d−1
n−1 e
Basis of powers
(Geramita ’96; Landsberg–T. ’10)
(Bialynicki-Birula–Schinzel ’08)
(Jelisiejew ’13)
−
n+d−7
n−3
(Ballico–De Paris ’13)
(Blekherman–T. ’14)
None of these is sharp.
25 / 25
Upper bounds for Waring rank
The vector space of polynomials has dimension
n+d−1
n−1
.
Upper bound for rank
n+d−1
n−1
n+d−1
−n+1
n−1
n+d−2
n−1
n+d−2
− n+d−6
n−1
n−3
n+d−2
− n+d−6
n−1
n−3
2d n1 n+d−1
n−1 e
Basis of powers
(Geramita ’96; Landsberg–T. ’10)
(Bialynicki-Birula–Schinzel ’08)
(Jelisiejew ’13)
−
n+d−7
n−3
(Ballico–De Paris ’13)
(Blekherman–T. ’14)
None of these is sharp.
25 / 25