Probabilistic algorithms for computing resolvent representations of regular differential ideals

Abstract

In a previous article [14], we proved the existence of resolvent representations for regular differential ideals. The present paper provides practical algorithms for computing such representations. We propose two different approaches. The first one uses differential characteristic decompositions whereas the second one proceeds by prolongation and algebraic elimination. Both constructions depend on the choice of a tuple over the differential base field and their success relies on the chosen tuple to be separating. The probabilistic aspect of the algorithms comes from this choice. To control it, we exhibit a family of tuples for which we can bound the probability that one of its element is separating.

A primitive element theorem for fields with commuting derivations and automorphisms

“…, α s ∈ Z 0 ). Theorem 1.1 and its improvements [31,33] have been used, for example, in algorithms and effective bounds for differential-algebraic equations [9,10,12,14,26], Galois theory of differential and difference equations [5,16], model theory of differential fields [25,38], control theory [3,13], and for connecting algebraic and analytic approaches to differential-algebraic equations [34,35,36].…”

A geometric index reduction method for implicit systems of differential algebraic equations

“…a univariate differential equation plus parameterizations of the variables) may be given by means of the notion of primitive element of an extension of differential fields. This construction, due to J. Ritt ([46], see also [49]), is known as a resolvent representation of the system Σ (see [6,7,11] for effective versions of it).…”

“…Suppose now that x i (j) is not an element of U and let p ji be its associated minimal polynomial in 1) as in (7). By taking the total derivative of p ji , we obtain the following relation:…”