Let $\mathrm{IMS}_n(m)$ count the $n\times n$ nonnegative integer matrices whose row sums, column sums, main-diagonal sum, and antidiagonal sum are all $m$. We determine the Ehrhart series $F_n(q)=\sum_{m\geq0}\mathrm{IMS}_n(m)q^m$ as reduced rational functions for $n=7$ and $n=8$. Their numerator--denominator degrees are respectively $(366,373)$ and $(540,548)$. Both numerators have positive integer coefficients, are palindromic and strictly unimodal. The proofs share one finite architecture: a signed SimpCone decomposition is evaluated by quotient characters over finite fields, a certified common denominator and Ehrhart reciprocity reduce the rational identity to finitely many coefficients, an explicit counting bound lifts modular congruences to integer equalities, and exact gcd computations prove reducedness. For order eight, a face-index pole certificate gives a degree-$598$ common denominator without enumerating the full face lattice, leaving $296$ independent coefficients in degrees $0$ through $295$. Once the candidate rational function is known, the first six production primes certify this finite prefix by the same bounded-coefficient argument used for order seven.
Let $C_n$ be a cyclic group of order $n$. We prove that if $(n,6)=1$, then every minimal zero-sum sequence of length four over $C_n$ has index one, thereby resolving the length-four index conjecture. After the gcd reduction, the nonunit case follows from the theorem of Shen-Xia-Li, and the remaining unit case is solved...
Hong-Jian Li, Ping-Zhi Yuan, Shijie Yuan et al.· 0 citations
Let $F$ be a rank-two symmetrizable Nahm sum, with any symmetrizer $\operatorname{diag}(d_1,d_2)$, a positive-definite rational matrix and rational shifts, and let $K$ be the field generated by its Nahm point; write $m=d_2/d_1$. We prove that if $q^cF$ is modular, then $[K:\mathbb{Q}]\le 24$. Modularity forces the high...
We construct an explicit Hermitian polynomial in six selfadjoint variables, $f=-1+\omega+\omega^*+M\sum_{j=1}^{15}(2-u_j-u_j^*)$, with integer coefficients, degree $72$, and exactly $33$ monomials. Here $\omega,u_1,\ldots,u_{15}$ are specified words, $*$ reverses words, and $M$ is a specified positive integer. Its norm...
For a positive integer $\alpha$, let $p_\alpha(n)$ denote the number of $\alpha$-colored partitions of $n$. The Rademacher-type expansion of Iskander, Jain, and Talvola is valid for every real $\alpha>0$, but when $\alpha>24$ it involves several polar terms and a two-parameter family of exponential sums $A_k^{(\alpha)}...
The Ehrhart series of the $n$th Birkhoff polytope is $\sum_{r\geq0}H_n(r)z^r$, where $H_n(r)$ counts nonnegative integer $n\times n$ matrices whose row and column sums all equal $r$. We present an exact method for computing this series using constant terms and finite fields. A root filter expresses $H_n(r)$ as a weight...
Xin-Ru Jiang, Guo-Ce Xin, Chen Zhang et al.· 0 citations
Let $\mathfrak{X}_k\subseteq{\sf Mat}_{m, n}$ denote the variety of $m\times n$ complex matrices with rank at most $k$. The power series and rational expressions for the Hilbert series of $\mathfrak{X}_k$ are known by geometric arguments, and equating these expressions yields a family of formulas generalizing the class...
Liam Buttitta, Ada Stelzer· 0 citations
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