An algebraically stable variety for a four-dimensional dynamical system reduced from the lattice super-KdV equation

TL;DR

This paper constructs an algebraically stable variety for a four-dimensional discrete dynamical system derived from the lattice super-KdV equation, proving quadratic growth of dynamical degree and identifying invariants.

Contribution

It introduces a rational variety that stabilizes the map algebraically and confirms the quadratic dynamical degree growth conjecture.

Findings

Proved the quadratic growth of the dynamical degree.

Constructed an algebraically stable variety for the system.

Identified invariants using Picard lattice action.

Abstract

In a prior paper the authors obtained a four-dimensional discrete integrable dynamical system by the traveling wave reduction from the lattice super-KdV equation in a case of finitely generated Grassmann algebra. The system is a coupling of a Quispel-Roberts-Thompson map and a linear map but does not satisfy the singularity confinement criterion. It was conjectured that the dynamical degree of this system grows quadratically. In this paper, constructing a rational variety where the system is lifted to an algebraically stable map and using the action of the map on the Picard lattice, we prove this conjecture. We also show that invariants can be found through the same technique.