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Characterization of invariants for equation (5.3) within finite order

Ascertain a finite and explicit set of independent invariants of the recurrence Un+g Sg+1(n) = Un+g+1 Sg+1(n+2) (equation (5.3)) by expressing Un+g+r Sg+r+2(n) solely in terms of (un, …, un+2g), and determine the resulting relationships among these invariants and the parameters (H1, …, Hg).

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Background

Invariants for the order-2g+1 recurrence (5.3) arise from expressions of the form un+g+r Sg+r+2(n). Due to their dependence on variables beyond the finite window (un,…,un+2g), establishing them as invariants of a fixed-order equation requires elimination using (5.3). The authors note that the existence and form of a finite set of independent invariants are not fully clarified, and a characterization is needed.

References

However, the following question remains unclear. The fact is that Un+g+r Str g+++2(n) depends on the variables (un,., Un+2g+2r) and therefore, generally speaking, un+g+rSgHr+2(n) should be brought to such a form it depends on the variables (un, ... , Un+2g), that is, it is necessary to express variables (Un+2g+1, ... , Un+2g+2r ) using (5.3). But what do we get as a result?

— Volterra map and related recurrences (2502.06908 - Svinin, 10 Feb 2025) in Section 5.3