We introduce an algebraic formulation of billiards on plane curves over algebraically closed fields, extending Glutsyuk’s complex billiards. Given an algebraic plane curve C of degree \(d \ge 2\) , algebraic billiards is a rational \((d-1)\) -to- \((d-1)\) surface correspondence on the space of unit tangent vectors based on C. We prove that the dynamical degree of the billiards correspondence is at most an explicit cubic algebraic integer \(\rho _d < 2d^2 - d - 3\) , depending on the degree d of C. As a corollary, for smooth real algebraic curves, the topological entropy of the classical billiards map is at most \(\log \rho _d\) . We further show that the billiards correspondence satisfies the singularity confinement property and preserves a natural 2-form. To prove our bounds, we construct a birational model that partially resolves the indeterminacy of algebraic billiards.
