The Erlangen program, introduced by Felix Klein in 1872, revolutionized geometry by redefining it as the study of properties of geometric spaces that remain invariant under a specified group of transformations. This perspective allowed different geometries, such as Euclidean, affine, and projective, to be understood and classified by the symmetries they preserve. In this framework, key geometric features, including those of curves and surfaces, are identified through invariants that do not change under the action of these groups. For curves, classical invariants include measures like arc-length and curvature, while for surfaces, Gaussian curvature is fundamental.

In real life, a geometric object is often discrete because it is determined by a finite set of specific data points or parameters, such as coordinate points or control points. In particular, a curve in an affine space can be determined by a discrete set of points. This discretization reflects how continuous geometric objects are often represented and manipulated computationally or experimentally.

We propose a complete description of invariants of a discrete affine curve under the action of the group of volume-preserving affine transformations. These invariants are local «in nature» and closely related to the corresponding differential invariants, providing a discrete analogue that captures the essential geometric properties preserved by such transformations.

The talk is based on a joint paper with Yaroslav Bazaikin.