Algorithms yield upper bounds in differential algebra

Abstract

Consider an algorithm computing in a differential field with several commuting derivations such that the only operations it performs with the elements of the field are arithmetic operations, differentiation, and zero testing. We show that, if the algorithm is guaranteed to terminate on every input, then there is a computable upper bound for the size of the output of the algorithm in terms of the size of the input. We also generalize this to algorithms working with models of good enough theories (including for example, difference fields).We then apply this to differential algebraic geometry to show that there exists a computable uniform upper bound for the number of components of any variety defined by a system of polynomial PDEs. We then use this bound to show the existence of a computable uniform upper bound for the elimination problem in systems of polynomial PDEs with delays.

Strong Consistency and Thomas Decomposition of Finite Difference Approximations to Systems of Partial Differential Equations

“…This discretization allows to solve numerically the last equations in the pressure-Poisson formulation when the pressure is determined from the Poisson pressure equation and the velocities from the momentum equations. Our numerical experiments with the two-dimensional analogue (39) of the new scheme have clearly demonstrated its superiority over the other two-dimensional schemes ( 40)- (42). In particular, the scheme reveals, at the discrete level, a surprisingly high accuracy preservation of the mass conservation law (continuity equation).…”

“…Since the difference Thomas decomposition partitions the solution space of the FDA, s-consistency holds if and only if every difference equation in each output subsystem approximates an element in the radical differential ideal generated by the elements in the input simple differential system. In the recent paper [39] it is argued that if a differential (or difference) decomposition algorithm terminates on every input, then one can provide a computable upper bound for the size of its output in terms of the input, i.e., an upper bound for number of output subsystems, their order and degree. Because of the termination of both decomposition algorithms, the upper bound estimation approach of paper [39] is applicable to differential and difference Thomas decompositions.…”

“…In the recent paper [39] it is argued that if a differential (or difference) decomposition algorithm terminates on every input, then one can provide a computable upper bound for the size of its output in terms of the input, i.e., an upper bound for number of output subsystems, their order and degree. Because of the termination of both decomposition algorithms, the upper bound estimation approach of paper [39] is applicable to differential and difference Thomas decompositions.…”

“…The first approximation, FDA1, given by Eqs. (39), was constructed in the paper [2] where its s-consistency was established. The difference approximation FDA4, given by Eqs.…”