Much of the existing literature on involutive bases concentrates on their efficient algorithmic construction. By contrast, we are here more concerned with their structural properties. Pommaret bases are particularly useful in this respect. We show how they may be applied for determining the Krull and the projective dimension, respectively, and the depth of a polynomial module. We use these results for simple proofs of Hironaka's criterion for Cohen-Macaulay modules and of the graded form of the Auslander-Buchsbaum formula, respectively.Special emphasis is put on the syzygy theory of Pommaret bases and its use for the construction of a free resolution. In the monomial case, the arising complex always possesses the structure of a differential algebra and it is possible to derive an explicit formula for the differential. Furthermore, in this case one can give a simple characterisation of those modules for which our resolution is minimal. These observations generalise results by Eliahou and Kervaire.Using our resolution, we show that the degree of the Pommaret basis with respect to the degree reverse lexicographic term order is just the Castelnuovo-Mumford regularity. This approach leads to new proofs for a number of characterisations of this regularity proposed in the literature. This includes in particular the criteria of Bayer/Stillman and Eisenbud/Goto, respectively. We also relate Pommaret bases to the recent work of Bermejo/Gimenez and Trung on computing the Castelnuovo-Mumford regularity via saturations.It is well-known that Pommaret bases do not always exist but only in so-called δ-regular coordinates. Fortunately, generic coordinates are δ-regular. We show that several classical results in commutative algebra, holding only generically, are true for these special coordinates. In particular, they are related to regular sequences, independent sets of variables, saturations and Noether normalisations. Many properties of the generic initial ideal hold also for the leading ideal of the Pommaret basis with respect to the degree reverse lexicographic term order. We further present

We present ecient algorithmic methods to detect Hopf bifurcation xed points in chemical reaction networks with symbolic rate constants, thereby yielding information about the oscillatory behavior of the networks. Our methods use the representations of the systems on convex coordinates that arise from stoichiometric network analysis. One of our methods then reduces the problem of determining the existence of Hopf bifurcation xed points to a rst-order formula over the ordered eld of the reals that can be solved using computational logic packages. The second method uses ideas from tropical geometry to formulate a more ecient method that is incomplete in theory but worked very well for the examples that we have attempted; we have shown it to be able to handle systems involving more than 20 species

We study the theory of systems with constraints from the point of view of the formal theory of partial differential equations. For finite-dimensional systems we show that the Dirac algorithm completes the equations of motion to an involutive system. We discuss the implications of this identification for field theories and argue that the involution analysis is more general and flexible than the Dirac approach. We also derive intrinsic expressions for the number of degrees of freedom.

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