Counting the number of lattice points of lattice polyhedra in a Euclidean space is a challenging problem in mathematics for long time. In 2-dimensional case, Pick’s Theorem is the most important and useful result relating the counting of lattice points of a lattice polygon with its area. An equivalent version of Pick’s Theorem says that the number of lattice points of a dilatation of a lattice polygon by a positive integer n is a quadratic polynomial of n. known as the Ehrhart polynomial of the lattice polygon. The coefficients of this quadratic polynomial can be considered as set function defined on the class of lattice polygons. It turns out that these set functions are finitely additive measures (known as valuations), invariant under unimodular transformations and translations. . In...[ Read more ]

Counting the number of lattice points of lattice polyhedra in a Euclidean space is a challenging problem in mathematics for long time. In 2-dimensional case, Pick’s Theorem is the most important and useful result relating the counting of lattice points of a lattice polygon with its area. An equivalent version of Pick’s Theorem says that the number of lattice points of a dilatation of a lattice polygon by a positive integer n is a quadratic polynomial of n. known as the Ehrhart polynomial of the lattice polygon. The coefficients of this quadratic polynomial can be considered as set function defined on the class of lattice polygons. It turns out that these set functions are finitely additive measures (known as valuations), invariant under unimodular transformations and translations. . In this thesis we study the vector space of invariant valuations on the class of lattice polygons of Euclidean plane. Our purpose is to classify such vector space by specifying a concrete basis. We proved that the coefficient valuations form a basis of the vector space of invariant valuations on lattice polygons.