2026/07/23 by Dun Qiu, Guoce Xin, Zihao Zhang
#math.CO
Let IMSn(m) denote the number of n× n nonnegative integer matrices whose row sums, column sums, and two main diagonal sums are all equal to m. We determine the Ehrhart series F7(q)=∑m≥ 0IMS7(m)qm as a reduced rational function. The denominator has degree 373 and cyclotomic factors of order at most 15; the numerator is a palindromic polynomial of degree 366 with nonnegative integer coefficients. Using the SimpCone decomposition, the associated polytope is represented as a sum of 166 million signed simplicial cones. The LRQC evaluator computes their generating functions over finite fields; a typical cone requires only one or two quotient characters, and the cost per character is nearly linear in the truncation degree T. An explicit common denominator together with Ehrhart reciprocity reduces the rational reconstruction to the prefix up to T=1256, while an explicit counting bound supplies the coefficient bounds needed for deterministic lifting from the prime fields to \mathbb Z. This prefix is independently computed for the whole cone family in eight prime fields. Exact Chinese remaindering lifts the verified residues to equality over \mathbb Z, the finite-prefix criterion proves the rational identity, and exact polynomial gcds prove that the displayed denominator is reduced.