Lattice points and Ehrhart polynomials

A lattice polytope in real d-space has all vertices in the integer lattice. Assume it is full dimensional. Its Ehrhart function L_P(t) counts lattice points in tP for nonnegative integers t. Ehrhart’s theorem says this count agrees with a degree-d polynomial, whose leading coefficient is ordinary Euclidean volume and whose constant coefficient is one. The value at zero counts the single point in the zero dilation, even if zero was not originally inside P.

The degree reflects dimension, not the number of vertices. For an r-dimensional lattice polytope in a larger ambient space, use relative Euclidean r-volume normalized so that a fundamental cell of the induced lattice has volume one. Some authors call r! times this quantity normalized volume; that convention must be divided by r! for the Ehrhart leading coefficient. The ambient d-volume is zero and is not the relevant leading coefficient. Integer dilation is also essential: the theorem does not claim one polynomial count for all real scale factors. The exact polynomial property can be established through triangulations and generating functions; we will use the theorem and derive elementary examples directly.

For positive integers t, Ehrhart–Macdonald reciprocity says the number of integer points in the interior of tP is (−1)^d times L_P(−t). The negative argument is polynomial evaluation, not the instruction to count a geometrically reflected dilation. In dimension two the sign is positive, and the linear coefficient of the Ehrhart polynomial is half the number of lattice points on the polygon’s boundary. Pick’s theorem gives a direct way to recover this formula from area and boundary data.

In a lattice polygon, the number of boundary lattice segments along an edge is the gcd of its two coordinate differences. Summing these edge gcds counts boundary points without double-counting vertices. Consequently L_P(t) equals area times t squared, plus half the original boundary count times t, plus one. Interior counts reverse the sign of the linear term. These formulas provide sensitive checks: a wrong boundary convention can leave the leading area term correct while corrupting every smaller dilation.

If the vertices are rational, a common denominator q makes qP a lattice polytope. The integer-dilate counting function is generally a quasipolynomial: one polynomial applies to each residue class of t modulo a period. A period divides a suitable common vertex denominator, although the minimal period can be smaller. It is therefore incorrect to insist that the denominator always equals the minimal period; period collapse is a genuine arithmetic phenomenon.

The simplest example is the interval from zero to one-half, whose count is floor(t/2) + 1. Even and odd t follow different linear formulas with the same leading slope. Conversely an invertible integer matrix with integer inverse preserves the lattice bijectively and leaves every dilation count unchanged. An arbitrary real shear or scaling preserves convex geometry but may move vertices off the lattice, so affine invariance of face counts does not imply affine invariance of Ehrhart counts.