co.combinatorics - Flipping Hilbert series of semigroup rings

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.

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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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