On a separation and irreducibility problem of polynomials arising from the nonlinear Schrodinger equation

We discuss an algebraic problem (Separation and Irreducibility Conjecture) which arises from the study of the nonlinear Schrodinger equation (NLS for short). This problem is about separation and irreducibility (over the ring of integers) of the characteristic polynomials of the graphs, describing blocks of a normal form for the NLS. For the cubic NLS the problem has been completely solved (see [C. Procesi, M. Procesi and B. Van Nguyen, The energy graph of the nonlinear Schrodinger equation, Rend. Lincei Mat. Appl. 24(2) (2013) 229-301]), meanwhile for higher degree NLS it is still open, even in small dimensions (see [C. Procesi, The energy graph of the non-linear Schrodinger equation, open problems. Int. J. Algebra Comput. 23(4) (2013) 943-962]). In this work, the author will give a partial answer for this problem, in particular, the author will prove Separation and Irreducibility Conjecture for NLS of arbitrary degree on one-dimensional and two-dimensional tori.