A polynomial moment approach to a rank condition for continuous-stage Runge--Kutta methods

In the study of energy-preserving methods for Hamiltonian systems, polynomial continuous-stage Runge--Kutta methods play an important role.

Necessary and sufficient conditions for such methods to be energy-preserving have already been established.

They are energy-preserving if the matrix $M\in \mathbb{R}^{s\times s}$ defining the method is symmetric, and the converse holds under the assumption that a certain $s\times \infty$ matrix $\Phi^\mathrm{CSRK}$ has full row rank.

It was conjectured in Remark 3 in Miyatake and Butcher (SIAM J.

Numer.

Anal., 2016) that the full-rank assumption should always hold for every consistent polynomial continuous-stage Runge--Kutta method.

In this paper, we prove the conjecture by showing that the matrix $\Phi^\mathrm{CSRK}$ has full row rank under the standard consistency condition.

The proof is a direct application of the polynomial moment problem solved by Pakovich and Muzychuk (Proc.

Lond.

Math.

Soc., 2009).