Abstract

We prove upper bounds on the order and degree of the polynomials involved in a resolvent representation of the prime differential ideal associated with a polynomial differential system for a particular class of ordinary first order algebraic-differential equations arising in control theory. We also exhibit a probabilistic algorithm which computes this resolvent representation within time polynomial in the natural syntactic parameters and the degree of a certain algebraic variety related to the input system. In addition, we give a probabilistic polynomial-time algorithm for the computation of the differential Hilbert function of the ideal.

Highlights

  • The notion of a resolvent representation of a prime differential ideal in a ring of differential polynomials was introduced by Ritt as a tool towards an algebraic elimination theory in the realm of differential equations, it can be traced back to the work of Kronecker

  • A resolvent representation of a prime differential ideal provides a parametrization of the generic zeros of the ideal by the general zeros of a single irreducible differential polynomial

  • This section is concerned with the notions of a primitive element of a differentially algebraic field extension and of a resolvent representation of a prime differential ideal introduced by Ritt

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Summary

Introduction

The notion of a resolvent representation of a prime differential ideal in a ring of differential polynomials was introduced by Ritt (see [28,27]) as a tool towards an algebraic elimination theory in the realm of differential equations, it can be traced back to the work of Kronecker (see [24]). We present a probabilistic algorithm for the computation of the differential Hilbert function of the ideal within complexity polynomial in n, m, r, d and linear in L (see Theorem 26), extending the results in [26] to positive-dimensional situations. The complexity results on the computation of characteristic sets in the differential setting given in [30] seem to yield single exponential complexity bounds for a probabilistic algorithm computing a resolvent representation (see [4]) for the specific systems we consider.A different approach to effective elimination over ordinary differential fields can be found in [15], where a general quantifier elimination procedure with doubly exponential complexity bounds is exhibited.

Differential algebra
Differential rings and fields
Rankings and characteristic sets
Data structures and algorithmic model
Generic algebraic-differential systems
Definitions and basic properties
Hilbert function and differential transcendence bases
Differential transcendence basis
The algorithms and their complexities
Resolvent representation
Existence of a primitive element and a resolvent representation
The minimal polynomial of a generic primitive element
The resolvent representation
Algorithmic computation of a resolvent representation
Computing the generic minimal polynomial
Computation of a primitive element
Computing a resolvent representation of the system
Over-determined differential systems
Independent equations
Extended resolvent representation
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