Generic moment-period conjecture

Determine whether, for every integer mgreater than or equal to 0 and outside a proper algebraic subset of choices of z in C^d, the coefficient functions c_{m,r}(t;z) in the real quasi-polynomial expansion of the mth discrete moment of a half-open integer parallelepiped have common smallest period equal to 1, so that complete period collapse does not occur.

Background

For a half-open integer parallelepiped Pi in Rd, the paper expresses the discrete moment sum over tPi intersected with Zd as a real quasi-polynomial in t through Barnes polynomials and polytope Dedekind sums. The coefficient functions can exhibit periods smaller than 1, and the authors contrast this behavior with complete period-collapse phenomena for closed rational polytopes.

The conjecture proposes that, for generic choices of the complex vector z, all coefficient functions associated with a fixed moment order share smallest period 1. It therefore asks whether the generic moment expansion avoids complete period collapse, while allowing exceptional algebraic choices of z to behave differently.

References

The following conjecture stands in sharp contrast to the complete period-collapse phenomena that occurs in the setting of closed rational polytopes. In the case of integral half-open parallelepipeds \Pi \subset Rd, it follows from Theorem \ref{thm:partial-alternating-differences} part \ref{part c of Ehrhart coefficients} that their quasi-coefficients all have a period of 1. This suggests the study of the smallest rational periods smaller than 1. For m\ge 0, define

\sum_{\bm p\in t\Pi\cap\mathbb Zd}\langle \bm p,\bm z\ranglem

\sum_{r=0}{d+m} c_{m,r}(t;\bm z)tr,

the real quasi-polynomial expansion obtained from the Barnes-polynomial and polytope Dedekind sum formula of Theorem \ref{thm: moments of dilated half-open parallelepipeds}. Then:

\begin{enumerate}[(a)] \item Outside a proper algebraic subset of choices of \bm z\in\mathbb Cd, the coefficient functions c_{m,r}(t;\bm z) have common smallest period equal to 1. In other words, there is no complete period collapse. \end{enumerate}

— Half-open integer parallelepipeds and polytope Dedekind sums  (2608.18408 - Robins et al., 19 Aug 2026) in Section 6, Further remarks; Conjecture (Generic moment-period version)

The following conjecture stands in sharp contrast to the complete period-collapse phenomena that occurs in the setting of closed rational polytopes. In the case of integral half-open parallelepipeds \Pi \subset Rd, it follows from Theorem \ref{thm:partial-alternating-differences} part \ref{part c of Ehrhart coefficients} that their quasi-coefficients all have a period of 1. This suggests the study of the smallest rational periods smaller than 1. For m\ge 0, define

\sum_{\bm p\in t\Pi\cap\mathbb Zd}\langle \bm p,\bm z\ranglem

\sum_{r=0}{d+m} c_{m,r}(t;\bm z)tr,

the real quasi-polynomial expansion obtained from the Barnes-polynomial and polytope Dedekind sum formula of Theorem \ref{thm: moments of dilated half-open parallelepipeds}. Then:

\begin{enumerate}[(a)] \item Outside a proper algebraic subset of choices of \bm z\in\mathbb Cd, the coefficient functions c_{m,r}(t;\bm z) have common smallest period equal to 1. In other words, there is no complete period collapse.

\item What is the smallest rational period of each coefficient c_{m,r}(t;\bm z)? \end{enumerate}

— Half-open integer parallelepipeds and polytope Dedekind sums  (2608.18408 - Robins et al., 19 Aug 2026) in Section 6, Further remarks; Conjecture (Generic moment-period version), part (b)