The predictable leading monomial property for polynomial vectors over a ring

The “predictable degree property”, a terminology introduced by Forney in 1970, is a property of polynomial matrices over a field F that has proven itself to be fundamentally useful for a range of applications. In this paper we strengthen this property into the “predictable leading monomial” property, and show that this PLM property is shared by minimal Gröbner bases for any positional term order (here: TOP and POT) in F[x]^q. The property is useful particularly for minimal interpolation-type problems. Because of the presence of zero divisors, minimal Gröbner bases over a finite ring of the type ℤ_pr (where p is a prime integer and r is an integer > 1) do not have the PLM property. We show how to construct, from an ordered minimal Gröbner basis, a so-called minimal Gröbner p-basis that does have a PLM property. The parametrization of all shortest linear recurrence relations of a finite sequence over ℤ_pr is a type of problem for which this is useful and we include an illustrative example.