# Hierarchical Zonotopal Spaces

Hierarchical Zonotopal Spaces

Suppose is an matrix. Let denote the hyperplane generated by a submatrix , with and denote by the set of all . The normal to a hyperplane is denoted by ; let . Then the central and the external ideals are generated by and , respectively. The semi-external ideal is generated by , where . The kernel of the central and external ideals are zonotopal spaces, while the kernel of the semi-external ideal is a hierarchical zonotopal space. This Demonstration shows the Hilbert series and Gröbner basis of the semi-external ideal .

X=(,...)

x

1

x

n

s×n

H

Y

Y⊆X

rank(Y)=s-1

ℱ(X)

H

Y

H

ℓ

H

m(H)=#{x∈X|x∉H}

(H):H∈ℱ(X)

m(H)

ℓ

H

:H∈ℱ(X)

m(H)+1

ℓ

H

I

:H∈ℱ(X)

m(H)+ϵ(H)

ℓ

H

ϵ(H)∈{0,1}

I

I