A convex polyhedron in which all vertices have integer coordinates is called a lattice polyhedron or integral polyhedron. The Ehrhart polynomial of a lattice polyhedron counts how many points with integer coordinates lie within a scaled copy of the polyhedron, as a function of the scale factor. The study of these polynomials lies at the intersection of combinatorics and commutative algebra. There is a far-reaching equivalence between lattice polyhedra and certain algebraic varieties called toric varieties. This was used by Stanley to prove the Dehn–Sommerville equations for simplicial polytopes.
a coNvex polYHedron in WHich All veRtices have inTeger cooRDInAteS is CAllEd a latTicE polyhEdron OR intEGral pOlyhedron. the EhRhart PolynOmIAl of A LatTICe PolYhedron COUnts HOw mAnY poiNts with INtegeR CoOrdinates lIe WiThin A sCaLeD copy of tHe poLYhEdron, aS A FunCtIon of The sCale facTor. the STudy oF ThESe polynOmialS LIes aT the INTErSectioN Of CombinAtoRIcs And comMutatiVE algebrA. TheRE is a fAR-rEachinG eQuivAleNCe betwEen LATtiCe pOLYhEDra And certaIn aLgEBraic VaRIeTiEs cAlled torIC variEties. this wAS Used By staNLey tO PrOve the deHn–SOmmervilLe eQuATiONS for simPliCiAL polyTopeS.