External zonotopal algebra
Nan Li (MIT)
Joint work with Amos Ron (UW-Madison)
Sep 24 2011, @ Wake Forest University
Outline
• From box splines to central zonotopal algebra
• External zonotopal algebra:
• focusing on a geometric interpretation of the duality.
Starting from the box splines
Given a list X of m vectors in Rd , consider the variable polytope
Π1X (t) = {y ∈ [0, 1]m | X · y = t}.
Define the box spline: Bx (t) = Vol Π1X (t). It is well known that Bx (t) is
a piece-wise polynomial in t = (t1 , . . . , td ), supported on the zonotope
Z (X ).
Example
Let X = { 10 , 01 , 11 }. Then the box spline
Bx (t) = Vol{(y1 , y2 , y3 ) ∈ [0, 1]3 | y1 10 + y2 01 + y3 11 =
t2 O


 −t2 + 2



t − t 

1
2

−t + 2
 +1  1 

 t2 − t1
t1 
+1

t2







o
(1, 0)
/
t1
t1
t2
} is
Starting from the box splines
Given a list X of m vectors in Rd , consider the variable polytope
Π1X (t) = {y ∈ [0, 1]m | X · y = t}.
Define the box spline: Bx (t) = Vol Π1X (t). It is well known that Bx (t) is
a piece-wise polynomial in t = (t1 , . . . , td ), supported on the zonotope
Z (X ).
Example
Let X = { 10 , 01 , 11 }. Then the box spline
Bx (t) = Vol{(y1 , y2 , y3 ) ∈ [0, 1]3 | y1 10 + y2 01 + y3 11 =
t2 O


 −t2 + 2



t − t 

1
2

−t + 2
 +1  1 

 t2 − t1
t1 
+1

t2







o
(1, 0)
/
t1
t1
t2
} is
D-space and J -ideal
Define the D-space to be the linear space in C[t1 , . . . , td ] generated by
all the polynomials in Bx (t) and their partial derivatives.
Example
In the previous example, we have
D(X ) = span {t1 , t2 , t1 −t2 +1, t2 −t1 +1, −t2 +2, −t1+2, 1} = span {1, t1 , t2 }.
There is an equivalent definition using inverse systems. Given a list of
vectors X , first define the ideal of cocircuits:
Y
J (X ) = Ideal{
ℓx | Z ⊂ X , Z ∩ B 6= ∅, ∀B ∈ B(X )}, then let
x∈Z
D(X ) = ker J (X ) := {g ∈ C[t1 , . . . , td ] | hf , g i = 0, ∀f ∈ J (X )},
where the inner product is defined as
∂
hf , g i = f
g |t=0 .
∂t
D-space and J -ideal
Define the D-space to be the linear space in C[t1 , . . . , td ] generated by
all the polynomials in Bx (t) and their partial derivatives.
Example
In the previous example, we have
D(X ) = span {t1 , t2 , t1 −t2 +1, t2 −t1 +1, −t2 +2, −t1+2, 1} = span {1, t1 , t2 }.
There is an equivalent definition using inverse systems. Given a list of
vectors X , first define the ideal of cocircuits:
Y
J (X ) = Ideal{
ℓx | Z ⊂ X , Z ∩ B 6= ∅, ∀B ∈ B(X )}, then let
x∈Z
D(X ) = ker J (X ) := {g ∈ C[t1 , . . . , td ] | hf , g i = 0, ∀f ∈ J (X )},
where the inner product is defined as
∂
hf , g i = f
g |t=0 .
∂t
D-space and J -ideal
Example
As in the previous examples, take X = { 10 , 01 , 11 }. Then
1
0
1
1
0
1
B(X ) =
,
,
,
,
,
.
0
1
0
1
1
1
So all the possible Z ’s such that Z ⊂ X , Z ∩ B 6= ∅, ∀B ∈ B(X ) are
1
0
1
1
0
1
1
0
1
,
,
,
,
,
and
,
,
.
0
1
0
1
1
1
0
1
1
Therefore,
J (X ) = Ideal{t1 t2 , t1 (t1 +t2 ), t2 (t1 +t2 ), t1 t2 (t1 +t2 )} = Ideal{t1 t2 , t12 , t22 },
and
D(X ) = ker J (X ) = span {1, t1 , t2 },
the same as computed by the definition from box splines.
Hyperplane arrangement and I-ideal
Example
For X = {
1
0
0 1
, 1 , 1 }, consider the hyperplane arrangement
?? O t2
??
??
??
H2
?? / t1
??
H3
H
1
where each vector in X defines the normal vector of one hyperplane.
Then define a power ideal
I = Ideal{t12 , t22 , (t1 − t2 )2 },
where
• each generator corresponds to a line in the hyperplane arrangement;
• and its power is the number of hyperplanes in the arrangement that
do not contain that line.
In this case I(X ) = Ideal{t12 , t22 , t1 t2 }.
I-ideal and P-space
From the power ideal I, define
P(X ) = ker I(X ).
In the above example, I = Ideal{t12 , t22 , t1 t2 }, so we have
P(X ) = span {1, t1, t2 }.
Theorem (de Boor-Dyn-Ron)
P(X ) = span {
Y
ℓx | Z ⊂ X , rank(X \Z ) = rank(X )}.
x∈Z
As in the example, for X = { 10 , 01 , 11 }, all the possible Z ’s are:
1
0
1
∅,
,
,
,
0
1
1
so
P(X ) = span {1, t1, t2 , t1 + t2 } = span {1, t1, t2 },
the same as we computed from the power ideal I.
I-ideal and P-space
From the power ideal I, define
P(X ) = ker I(X ).
In the above example, I = Ideal{t12 , t22 , t1 t2 }, so we have
P(X ) = span {1, t1, t2 }.
Theorem (de Boor-Dyn-Ron)
P(X ) = span {
Y
ℓx | Z ⊂ X , rank(X \Z ) = rank(X )}.
x∈Z
As in the example, for X = { 10 , 01 , 11 }, all the possible Z ’s are:
1
0
1
∅,
,
,
,
0
1
1
so
P(X ) = span {1, t1, t2 , t1 + t2 } = span {1, t1, t2 },
the same as we computed from the power ideal I.
Central zonotopal algebra
Given a list a vectors X , construct
• group one: cocircuit ideal J → D-space;
• group two: power ideal I → P-space.
• The two groups are dual to each other. In particular, the two spaces
D and P have the same Hilbert function.
Central zonotopal algebra
• is not an algebra;
• is the idea of connecting a matriod with the geometry (zonotopes,
hyperplane arrangement) and the four algebraic objects.
External zonotopal algebra
Given a list of vectors X , append an extra list of vectors Y to X , and
consider some collection of bases B′ ⊂ B(X ∪ Y ). Now define
Y
J B′ = Ideal{
ℓx | Z ⊂ X ∪ Y , Z ∩ B 6= ∅, ∀B ∈ B′ }.
x∈Z
and
D B′ = ker J B′ .
Remark
• Introducing this extra Y will help us understand X better;
• the definition of D B′ depends on Y , but the properties (e.g.
dimension and Hilbert function) we are interested in will be
independent of Y .
Question
• In general, we have dim DB′ ≥ #B′ (de Boor-Ron). When do we
have the equality (B′ is coherent)?
• How to define the corresponding P-space, which is dual to the
D-space?
External zonotopal algebra
Given a list of vectors X , append an extra list of vectors Y to X , and
consider some collection of bases B′ ⊂ B(X ∪ Y ). Now define
Y
J B′ = Ideal{
ℓx | Z ⊂ X ∪ Y , Z ∩ B 6= ∅, ∀B ∈ B′ }.
x∈Z
and
D B′ = ker J B′ .
Remark
• Introducing this extra Y will help us understand X better;
• the definition of D B′ depends on Y , but the properties (e.g.
dimension and Hilbert function) we are interested in will be
independent of Y .
Question
• In general, we have dim DB′ ≥ #B′ (de Boor-Ron). When do we
have the equality (B′ is coherent)?
• How to define the corresponding P-space, which is dual to the
D-space?
Coherency
It is known that in the following cases, B′ is coherent, i.e.,
dim DB′ = #B′ .
• central case, by Holtz-Ron;
• external case, by Holtz-Ron;
• semi-external, by Holtz-Ron-Xu.
Our set up:
Assume that Y = {y1 , y2 , . . .} contains sufficiently many vectors in
general position in X ∪ Y . Consider a function κ : 2X → N. We consider
a class of B′ = Bκ depending κ.
Theorem (L.-Ron)
dim DBκ = #Bκ if and only if κ is solid, i.e., span Z ⊂ span Z ′ implies
κ(Z ) ≤ κ(Z ′ ), for any Z , Z ′ ⊂ X .
Remark
We can pick special κ’s to recover the above three cases.
Approaching the second question: family of P-spaces
It is known that the P-spaces defined as follows are dual to their
corresponding D-spaces:
• central case, by Holtz-Ron;
• external case, by Holtz-Ron, Postnikov-Shapiro-Shapiro;
• semi-external, by Holtz-Ron-Xu.
There are some recent development of external P-spaces from another
approach without introducing the D-spaces. They studied the Hilbert
functions of the P-spaces and their connection to the power ideals.
• (Ardila-Postnikov) For k ≥ 0, define
X Y
P k (X ) =
ℓ x Πk ,
Z ⊂X x∈X \Z
where Πk is the set of polynomials in d variables with degree at
most k.
• (Lenz) Generalized the results by Holtz-Ron-Xu and
Ardila-Postnikov.
Defining a unified P-space
Here, we generalize the above P-spaces using a solid assignment
κ : 2X → N,
X Y
P κ :=
ℓx Πκ(Z ) ,
Z ⊂X x∈X \Z
where Πκ(Z ) is the set of polynomials in d variables with degree at most
κ(Z ).
Theorem (L.-Ron)
P κ is dual to D κ if and only if κ is incremental, i.e., for every Z ⊂ X
and x ∈ X , we have κ(Z ∪ x) ≤ κ(Z ) + 1.
• Assume the assignment κ to be solid and incremental. We studied
the homogeneous basis for P κ , and its Hilbert function.
Defining a unified P-space
Here, we generalize the above P-spaces using a solid assignment
κ : 2X → N,
X Y
P κ :=
ℓx Πκ(Z ) ,
Z ⊂X x∈X \Z
where Πκ(Z ) is the set of polynomials in d variables with degree at most
κ(Z ).
Theorem (L.-Ron)
P κ is dual to D κ if and only if κ is incremental, i.e., for every Z ⊂ X
and x ∈ X , we have κ(Z ∪ x) ≤ κ(Z ) + 1.
• Assume the assignment κ to be solid and incremental. We studied
the homogeneous basis for P κ , and its Hilbert function.
Geometric interpretation of the
duality
Given a list of vectors X , for example, X = { 10 , 01 }. Append a list
Y = {y1 , y2 , y3 } to X and consider the corresponding generic affine
hyperplane arrangement.
y2
x1
?
?
?
?
?
?





i y3
?
i i
i ?i? i
i
?
i i 
i i 
?
x2
i
i
?

?

? y

1
Geometric interpretation of the
duality
Given a list of vectors X , for example, X = { 10 , 01 }. Append a list
Y = {y1 , y2 , y3 } to X and consider the corresponding generic affine
hyperplane arrangement.
y2
x1
?
?
?
?






i y3
?
i i
i ?i? i
i
?
i i 
i i 
x2
i
•? ?
i

?

? y

1
?
?
Each basis B ∈ B(X ∪ Y ) defines a vertex v(B) ∈ Rn , viz, the common
zero of the polynomials (qx )x∈B . For example, {x2 , y1 } defines the black
vertex.
Geometric interpretation of the
duality
Given a list of vectors X , for example, X = { 10 , 01 }. Append a list
Y = {y1 , y2 , y3 } to X and consider the corresponding generic affine
hyperplane arrangement.
y2
x1
?
?
•?
?






i y3
?
i i
i•?•i? i
i
?
i i •
i i•• • 
x2
i
•
•? ?
i

?
•

? y

1
?
?
Since the arrangement is generic, there is a bijection between B(X ∪ Y )
and the set of all the vertices (0-dim intersections) of the hyperplane
arrangement.
Geometric interpretation of the
duality
Given a list of vectors X , for example, X = { 10 , 01 }. Append a list
Y = {y1 , y2 , y3 } to X and consider the corresponding generic affine
hyperplane arrangement.
y2
x1
?
?
•?
?






i y3
?
i i
i•?•i? i
i
?
i i •
i i◦• • 
x2
i
◦
•? ?
i

?
•

? y

1
?
?
Now in our set up, we choose some Bκ ⊂ B(X ∪ Y ), which corresponds
to a subset of vertices v(Bκ ).
Geometric interpretation of the
duality
Given a list of vectors X , for example, X = { 10 , 01 }. Append a list
Y = {y1 , y2 , y3 } to X and consider the corresponding generic affine
hyperplane arrangement.
y2
x1
?
?
•?
?






i y3
?
i i
i•?•i? i
i
?
i i •
i i◦• • 
x2
i
◦
•? ?
i

?
•

? y

1
?
?
Recall that for a solid and incremental κ, we have
#v(Bκ ) = #Bκ = dim Pκ = dim Dκ .
We will build connections from the above points set v(Bκ ) to both
D κ -space and P κ -space.
Geometric interpretation of the duality
For a solid and incremental κ, let v(Bκ ) be the point set of the vertices
in the hyperplane arrangement corresponding to Bκ .
Theorem (L.-Ron)
• Π(v(Bκ )) = Dκ , where Π(v(Bκ )) is the least space of the set
v(Bκ ), where the least space is a polynomial space assigned to a
point set, whose dimension is the cardinality of the point set;
• there exists a Lagrange basis of Pκ = span {qB | B ∈ Bκ }
corresponding to the set v(Bκ ). Namely, for each v(B) ∈ v(Bκ ),
qB (v(B)) 6= 0 and qB (v(B ′ )) = 0 for all B ′ 6= B.
t2O


T A K
 H  N 


Y U
O 
/t
o (1,0)
1
Geometric interpretation of the duality
For a solid and incremental κ, let v(Bκ ) be the point set of the vertices
in the hyperplane arrangement corresponding to Bκ .
Theorem (L.-Ron)
• Π(v(Bκ )) = Dκ , where Π(v(Bκ )) is the least space of the set
v(Bκ ), where the least space is a polynomial space assigned to a
point set, whose dimension is the cardinality of the point set;
• there exists a Lagrange basis of Pκ = span {qB | B ∈ Bκ }
corresponding to the set v(Bκ ). Namely, for each v(B) ∈ v(Bκ ),
qB (v(B)) 6= 0 and qB (v(B ′ )) = 0 for all B ′ 6= B.
t2O


T A K
 H  N 


Y U
O 
/t
o (1,0)
1