Two families of Hermite normal form simplices

Source abstract

We investigate the Ehrhart coefficients and the integer decomposition property for two families of Hermite normal form simplices studied by Bruckamp, Caicedo, and Juhnke. The first family consists of simplices of the form $S_{\boldsymbol{a}}=\mathrm{conv}(0,e_1,\ldots,e_{d-1},\boldsymbol{a})$, where $\boldsymbol{a}=(a_1,\ldots,a_{d-1},N)$, while the second comprises the simplices $T_{d,N}=\mathrm{conv}\bigl(0,e_1,\ldots,e_{d-2},(d-2,\ldots,d-2,d-1,0),(1,\ldots,1,N)\bigr)$. For the family $T_{d,N}$, we establish the unimodality of the Ehrhart coefficients in arbitrary dimensions and completely classify the log-concave and real-rooted cases. For the sub-family $S_{\boldsymbol{a}}$ with $\boldsymbol{a}=(N-q,\ldots,N-q,N)$, we characterize both the integer decomposition property and the existence of a regular unimodular triangulation via a congruence condition on a negative continued fraction. More generally, we extend our analysis of the integer decomposition property and unimodular triangulations to $S_{\boldsymbol{a}}$ for arbitrary vectors $\boldsymbol{a}$. As a consequence, our results resolve three open problems posed by Bruckamp, Caicedo, and Juhnke.