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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 $1$?

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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
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2 Answers

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.

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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.)

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