Involutive bases are a special kind of Gröbner bases with additional combinatorial properties that make them very useful for many applications. They exist in many polynomial algebras (also non-commutative ones) including ordinary polynomials and linear differential or difference operators. The project was divided into three parts: fundamental theory, algorithms and implementations, and applications.
On the theoretical side, numerous results on the relations between different kinds of involutive bases, Gröbner bases and characteristic sets have been obtained both for ordinary and for differential ideals. Several characterisation theorems for involutive bases have been proved and the computation of (differential) dimension polynomials has been studied. We have thoroughly investigated the homological approach to involution via Spencer cohomology.
The involutive theory provides a highly competitive algorithm for the construction of Gröbner bases. This was shown with extensive benchmarks in comparison with the classical Buchberger approach. Within the project several implementations in different computer algebra systems (Mathematica, Maple, MuPAD, Reduce) as well as in stand-alone C/C++ programs have been developed. Furthermore, many optimisations have been studied theoretically and practically. An algebraic algorithm for the geometric completion to involution was developed (including a constructive solution of the problem of (-regularity). Finally, we studied the parallelisation of completion algorithms.
Applications in symmetry theory, constrained dynamics and numerical analysis have been investigated. We performed a complete group classification of the Navier-Stokes equations for a compressible viscous heat-conducting gas with respect to five independent functions. Furthermore, an invariant description of pattern formation conditions in biological reaction-diffusion equations was given. We demonstrated that completion to involution lies at the heart of constrained dynamics and studied several concrete physical models like SU(2) Yang-Mills mechanics. Finally, we showed that the theory of involutive systems provides a very natural framework for studying the numerical integration of overdetermined systems. Many index concepts developed by numerical analysts possess a simple interpretation in this theory. Furthermore, obstructions to involution may become integrability conditions in a semi-discretisation.
Impact:Involution is a key concept for under- or overdetermined systems of differential or algebraic equations which are ubiquitous in engineering or natural sciences; for example, all fundamental interactions in modern physics lead to such systems. Within the project significant progress has been made towards a better theoretical understanding of involution, a number of highly efficient implementations in different environments have been provided and the high potential of involutive techniques in applications in many different fields has been demonstrated.
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