Werner M. Seiler

A combinatorial approach to involution and δ -regularity I: involutive bases in polynomial algebras of solvable type

Involutive bases are a special form of non-reduced Gröbner bases with additional combinatorial properties. Their origin lies in the Janet–Riquier theory of linear systems of partial differential equations. We study them for a rather general class of polynomial algebras including also non-commutative algebras like those generated by linear differential and difference operators or universal enveloping algebras of (finite-dimensional) Lie algebras. We review their basic properties using the novel concept of a weak involutive basis and present concrete algorithms for their construction. As new original results, we develop a theory for involutive bases with respect to semigroup orders (as they appear in local computations) and over coefficient rings, respectively. In both cases it turns out that generally only weak involutive bases exist.

A combinatorial approach to involution and δ -regularity II: structure analysis of polynomial modules with pommaret bases

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 which is generically minimal for componentwise linear modules. 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. Here a minimal resolution is obtained, if and only if a stable module is treated. 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 always the Castelnuovo–Mumford regularity. This approach leads to new proofs for a number of characterisations of this invariant 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. 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, although the latter one is in general not Borel-fixed. We present a deterministic approach for the effective construction of δ-regular coordinates that is more efficient than all methods proposed in the literature so far.

On the arbitrariness of the general solution of an involutive partial differential equation

The relationship between the strength of a differential equation as introduced by Einstein, its Cartan characters, and its Hilbert polynomial is studied. Using the framework of formal theory previous results are extended to nonlinear equations of arbitrary order and to overdetermined systems. The problem of computing the number of arbitrary functions in the general solution is treated. Finally, the effect of gauge symmetries is considered.

Involution and constrained dynamics. I. The Dirac approach

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.

Algorithmic Methods for Lie Pseudogroups

We present an algorithm to complete any given system of differential equations to an involutive system as needed e.g. for concrete applications of Lie pseudogroups. It is based on jet bundle formalism and formal theory. An implementation in the computer algebra system AXIOM is described.

An Efficient Algebraic Algorithm for the Geometric Completion to Involution

Abstract. We describe an adaption of a differential algebraic completion algorithm for linear systems of partial differential equations that allows us to deduce intrinsic differential geometric information like the number of prolongations and projections needed for the completion. This new hybrid algorithm represents a much more efficient realisation of the classical Cartan–Kuranishi completion than previous purely geometric ones.A classical problem in geometric completion theory is the existence of δ-singular coordinate systems in which the algorithms do not terminate. We develop a new and a very simple criterion for δ-singularity based on a comparison of the Janet and the Pommaret division. This criterion can also be used for the direct construction of δ-regular coordinates.

Detection of Hopf bifurcations in chemical reaction networks using convex coordinates

We present efficient algorithmic methods to detect Hopf bifurcation fixed 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 fixed points to a first-order formula over the ordered field of the reals that can be solved using computational logic packages. The second method uses ideas from tropical geometry to formulate a more efficient 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.

Numerical integration of constrained Hamiltonian systems using Dirac brackets

We study the numerical properties of the equations of motion of constrained systems derived with Dirac brackets. This formulation is compared with one based on the extended Hamiltonian. As concrete examples, a pendulum in Cartesian coordinates and a chain molecule are treated.

INVOLUTION AND CONSTRAINED DYNAMICS. II: THE FADDEEV-JACKIW APPROACH

For pt.I see Seiler et al., ibid., vol.28., p.4431 (1995). We study the symplectic approach to first-order systems with constraints from the point of view of the formal theory of differential equations. We concentrate especially on systems without first-class constraints and give a geometric interpretation of an approach recently proposed by Barcelos-Neto and Wotzasek (1992). We further study the numerical properties of this approach. We also comment on some problems concerning the application to field theories.

DIFFERENTIAL EQUATIONS, SPENCER COHOMOLOGY, AND COMPUTING RESOLUTIONS

Abstract We propose a new point of view of the Spencer cohomology appearing in the formal theory of differential equations based on a dual approach via comodules. It allows us to relate the Spencer cohomology with standard constructions in homological algebra and, in particular, to express it as a Cotor. We discuss concrete methods for its construction based on homological perturbation theory.