On the Coefficients of the Ehrhart Tensor Polynomials

The term of Ehrhart tensor polynomials is a natural popularization of the Ehrhart polynomial of lattice polytopes which is a much more used in different applications. As our fundamental instrument, we present $\mathrm{h}^{\mathrm{r}}$-tensor polynomials which expand the idea of the Ehrhart $\mathrm{h}*$-polynomial for a polytope, and networks. It also examines their coefficients for positive semi definiteness rather than the standard $\mathrm{h}*-$ polynomial. This paper gives the proof of two theorems that compute the coefficients of the Ehrhart Tensor polynomials (h
³
and $\mathrm{h}^{4}$). The positivity for these coefficients is also proved. Some applications on the Ehrhart tensor polynomials are given.