New order bounds in differential elimination algorithms

In the case of one derivation, such a bound was given by Golubitsky et al. (2008). The only known bound in the case of several derivations was given by the authors of the present paper in 2016. The bound was achieved by associating to the algorithm antichain sequences whose lengths can be bounded using the results of León Sánchez and Ovchinnikov (2016). In the present paper, the above result by the current authors is generalized and significantly improved.