On the differentiation index of generic DAE systems

LISI D'ALFONSO; GABRIELA JERONIMO; PABLO SOLERNÓ

Madrid, España

Congreso; International Congress of Mathematicias 2006; 2006

International Mathematical Union

The differentiation index is a well known invariant associated to a DAE (differential algebraic equation) system. Roughly speaking, this invariant equals the minimum number of times that a given DAE system must be differentiated in order to determine the derivatives of the unknowns as continuous functions of the unknowns themselves. In some sense, it represents a measure of the complexity of the system from the point of view of its resolution. We give a precise algebraic definition of the differentiationindex for DAE systems of arbitrary order with generic second members by means of the study of stationary properties of the ranks of suitable associated Jacobian submatrices. Our approach induces a polynomial time algorithm for the computation of the differentiation index. Also, it allows us to show an upper bound for the differentiation index in terms of the orders of the given equations, similar to the one that can be obtained by rewriting methods. In addition, we show another equivalent definition of the index in terms of a filtration given by the successive differentiation of the input equations. As a by-product, we obtain an upper bound for the regularity of the Hilbert-Kolchin function associated to the DAE systems under consideration not depending on characteristic sets. We also establish some quantitative and algorithmic results concerning differential transcendence bases for first order DAE systems, and we are able to give a simple description of these systems distinguishing the variables by their interrelations.