The symplectic characteristic polynomial

TL;DR

This paper introduces the symplectic characteristic polynomial, a new invariant in symplectic linear algebra, and proves its completeness for symplectically diagonalizable endomorphisms.

Contribution

It defines the symplectic characteristic polynomial and establishes it as a complete invariant for a class of symplectic endomorphisms.

Findings

The symplectic characteristic polynomial generalizes the classical characteristic polynomial.

It is a complete invariant for symplectically diagonalizable endomorphisms.

The polynomial is in two variables and captures symplectic structure information.

Abstract

We introduce the notion of the symplectic characteristic polynomial of an endomorphism of a symplectic vector space. This is a polynomial in two variables and can be considered as a generalization of the characteristic polynomial of the endomorphism in the context of symplectic linear algebra. One of the goal of this paper is to prove that the symplectic characteristic polynomial is a complete symplectic invariant of symplectically diagonalizable endomorphisms.