MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I'll first give intuition, and then give a precise statement.

For $|z|<1$, we have $\sum_{i \geq 0} z^i = 1/(1-z)$. For $|z|>1$, we have $\sum_{i<0} (-1) z^i=1/(1-z)$. Thus, the two functions

$\phi(i) = \begin{cases} 1 \quad i \geq 0 \\ 0 \quad i<0 \end{cases}$ and $\psi(i) = \begin{cases} 0 \quad i \geq 0 \\ -1 \quad i<0 \end{cases}$

have the "same" Fourier transform. This question is about generalizations of this phenomenon to higher dimensions.

Let $\phi : \mathbb{Z}^n \to \mathbb{R}$ be a function such that (1) $\phi$ can be written as a rational combination of the characteristic functions of finitely many rational cones and (2) there is a linear function $\lambda$ so that $\phi$ vanishes on $\{ e : \lambda(e) \leq 0 \} \setminus \{ 0 \}$. We define $$h(\phi) = \sum \phi(e) z^e.$$ This sum converges somewhere, and gives a rational function of $z$.

Given $\phi$ as above, and given a generic linear functional $\zeta$, one can show that there is a unique function $\phi^{\zeta}$ such that (1) $\phi$ can be written as a rational combination of the characteristic functions of finitely many cones (2) $\phi$ vanishes on $\{ e : \zeta(e) \leq 0 \} \setminus \{ 0 \}$ and (3) $h(\phi^{\zeta}) = h(\phi)$.

For example, the above computation shows that $\phi^{(-1)} = \psi$. If $\phi$ is the characteristic function of a simplicial cone, this operation is easy to describe. In principle, this means we can always calculate $\phi^{\zeta}$ by writing $\phi$ as a linear combination of simplicial cones.

If $\phi$ is the characteristic function of a cone, what is known about $\phi^{\zeta}$? For example, is it true that (a) all the values of $\phi^{\zeta}$ have the same sign (NO, counter-example below), or that (b) $\phi^{\zeta}$ only takes the values $-1$, $0$ and $1$?

share|cite|improve this question
I tagged this ac.commutative-algebra because I have a hunch this is related to local cohomology of semigroup rings, but I don't know a precise connection. – David Speyer Jan 25 '10 at 17:32

Counter-example to (a). Consider the cone spanned by $(0,0,1)$, $(1,0,1)$, $(0,1,1)$ and $(1,1,1)$. This has generating function $$\frac{1-xyz^2}{(1-z)(1-xz)(1-yz)(1-xyz)}.$$

Let $\zeta$ be $(2,0,-1)$. If I have not made any errors, then the unique representation of this rational generating function using exponents in the half plane $\{ (i,j,k) : 2i \geq k \}$ is $$- \frac{x}{(1-x)(1-xz)(1-xyz)} + \frac{y^{-1} z^{-2}}{(1-x)(1-z^{-1})(1-y^{-1}z^{-1})}.$$

I believe that the corresponding two simplicial cones don't overlap, so the function $\phi^{\zeta}$ is $-1$ on the first cone and $1$ on the second.

share|cite|improve this answer

Alex Fink and I prove a number of lemmas about this operation in section 6 of our paper on matroids and equivariant localization. (This paper was still being written at the time I asked this question.)

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.