Frumkin, M. A. On the number of nonnegative integer solutions of a system of linear Diophantine equations. (English) Zbl 0477.90048 Ann. Discrete Math. 11, 95-108 (1981). Two assertions are proved about the number \(\nu(k)\) of integer points in the polyhedron \[ a_1x_1+\dots+a_nx_n = k,\quad a_1, \dots, a_n,\, k\in\mathbb Z^n,\;x_1, \dots, x_n\in\mathbb R_+\,. \]Theorem 1. There exists a partition \(\mathbb R^n=\bigcup_{i=1}^N C_i\), where \(C_i\) is a cone with integer generators and with the vertex in 0 such that \(\nu(k)|_{C_i\cap\mathbb Z^n}\) is either infinity or a quasipolynomial.Theorem 2. If in addition the matrix \((a_1, \dots, a_n)\) is totally unimodular, then \(\nu(k)|_{C_i\cap\mathbb Z^n}\) is either infinity or a polynomial.For the entire collection see [Zbl 0465.00007]. Reviewer: M. Frumkin Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page MSC: 11D04 Linear Diophantine equations 90C10 Integer programming Keywords:number of nonnegative integer solutions; system of linear Diophantine equations; quasipolynomial; polynomial; totally unimodular matrix PDFBibTeX XMLCite \textit{M. A. Frumkin}, Ann. Discrete Math. 11, 95--108 (1981; Zbl 0477.90048) Full Text: DOI