An Efficient Algebraic Algorithm for the Geometric Completion to Involution

Overdetermined Elliptic Systems

“…Since we study only linear systems, we emphasize the algebraic side and briefly describe the construction of involutive bases for linear differential systems [17]. More details and the precise connection of these bases to the formal theory can be found in [21]; for a general introduction to involutive bases, see [9], [46]. Janet introduced the fundamental concept of multiplicative variables: we assign to each equation in the system a subset of the set of all independent variables as its multiplicative variables.…”

“…Furthermore, we ignore here the problem of δ-regularity (which concerns the termination of the described completion algorithm in certain "bad" coordinate systems), as it is related to characteristics and thus of minor importance for elliptic systems. Details (and a constructive solution) are contained in [21].…”

“…which is elliptic if and only if the polynomial (21) has no real zeros. Thus, we have transformed the operator A to an equivalent operator A (2) , which is elliptic if and only if the operator A is DN-elliptic.…”

On the Numerical Analysis of Overdetermined Linear Partial Differential Systems

“…Because of our assumption on rank C n , it is not difficult to see that if such matrices H i , K exist, they are uniquely determined by (10). We derive the compatibility conditions of the linear system (9) under the assumption that it is involutive.…”

“…We present now a completion algorithm for linear systems that combines algebraic and geometric ideas. More details on the algorithm can be found in [10]; an implementation in the computer algebra system MuPAD is briefly described in [1]. In order to formulate our algorithm, we need some further notations.…”