Hierarchical Zonotopal Spaces

central to external ℐ ideal

ϵ

0

1

2

3

4

5

6

repeat the columns in the matrix X

r

1

2

3

4

5

r

2

1

2

3

4

5

r

3

1

2

3

4

5

r

4

1

2

3

4

5

The matrix input X is

1	0	0	1
0	1	0	1
0	0	1	1

.

The semi-external ideal I is generated by 

2

x

1

,

2

x

2

,

2

x

3

,

2

(

x

1

-

x

2

)

,

2

(

x

1

-

x

3

)

,

2

(

x

2

-

x

3

)

.

The Hilbert series of I is 3x+1.

The Gröbner basis of I is



2

x

3

,

x

2

x

3

,

2

x

2

,

x

1

x

3

,

x

1

x

2

,

2

x

1

.

Suppose

X=(

x

1

,...

x

n

)

is an

s×n

matrix. Let

H

Y

denote the hyperplane generated by a submatrix

Y⊆X

, with

rank(Y)=s-1

and denote by

ℱ(X)

the set of all

H

Y

. The normal to a hyperplane

H

is denoted by

ℓ

H

; let

m(H)=#{x∈X|x∉H}

. Then the central and the external ideals are generated by



m(H)

ℓ

H

(H):H∈ℱ(X)

and



m(H)+1

ℓ

H

:H∈ℱ(X)

, respectively. The semi-external ideal

I

is generated by



m(H)+ϵ(H)

ℓ

H

:H∈ℱ(X)

, where

ϵ(H)∈{0,1}

. The kernel of the central and external ideals are zonotopal spaces, while the kernel of the semi-external ideal

I

is a hierarchical zonotopal space. This Demonstration shows the Hilbert series and Gröbner basis of the semi-external ideal

I

.