Degree and Dimension Estimates for Invariant Ideals of \(P\) -Solvable Recurrences

Motivated by the generation of polynomial loop invariants of computer programs, we study \(P\)-solvable recurrences. While these recurrences may contain non-linear terms, we show that the solutions of any such relation can be obtained by solving a system of linear recurrences. We also study invariant ideals of \(P\)-solvable recurrences (or equivalently of while loops with no branches). We establish sharp degree and dimension estimates of those invariant ideals.