Differential Elimination Theory (DET)

Differential equations allow us to model a great variety of phenomena arising in physics, engineering, biology, financial analysis etc. Such ordinary and partial differential equations (ODEs and PDEs) have been studied for a long time, and there are both symbolic and numerical approaches to the analysis and solution of differential equations. In this project we are mainly concerned with the symbolic treatment of ODEs and PDEs, i.e. reductions of differential problems to algebraic ones and algebraic methods for these derived problems. This idea of an algebraic approach to differential equations has a long history. In the 19th century S. Lie initiated the investigation of transformations, which leave a given differential equation invariant. Such transformations are commonly known as Lie symmetries. They form a group, a so-called Lie group. The basic idea here is to find a group of symmetries of the differential equations and then use this group to reduce the order or the number of variables appearing in the equation. At the beginning of the 20th century Riquier and Janet introduced the concept of involutivity for systems of algebraic differential equations. Their theory leads to canonical systems of generators for differential ideals, and the algorithm for generating such canonical Janet bases is strikingly similar to the method of Gröbner bases for generating canonical systems for algebraic ideals devoloped by Buchberger. Another approach to symbolic solution of differential equations has been initiated by J. Liouville for the study of formal integration, and Liouville`s method has been extended to an algorithm for solving linear differential equations. There are several advantages of an algebraic analysis of differential equations: sometimes we are actually able to go the whole way and find a symbolic solution to the given DE; if we can derive symmetries of a DE, then this helps a lot in verifying numerical schemes for approximation of solutions; we can decide whether a system of ODEs or PDEs admits a solution; if there are solutions, then we can derive differential systems in triangular form such that the solutions of the original system are the (non-singular) solutions of the output system; we can get a complete overview of the algebraic relations satisfied by the solutions of a given system of differential algebraic equations (DAEs). An effective treatment of these problems depends crucially on algorithms in differential elimination theory, i.e. the theory of differential ideals in a ring of differential polynomials. The algebraic theory of elimination is well developed, and we can answer the main questions such as consistency of ideals, equality of ideals, the membership problem, the radical membership problem, the arithmetic of ideals, etc. constructively. The methods for solving these problems are resultants, Gröbner bases, and algebraic involutive bases. For differential ideals there are still many open problems. For instance, the membership problem or the ideal inclusion problems for finitely generated differential ideals are still not solved. The analysis and improvement of existing algorithms as well as the development of new algorithmic methods in differential elimination theory is the major goal of the project. Additionally, we will also consider the important application area of control theory.

Differential equations allow us to model a great variety of phenomena arising in physics, engineering, biology, financial analysis etc. Such ordinary and partial differential equations (ODEs and PDEs) have been studied for a long time, and there are both symbolic and numerical approaches to the analysis and solution of differential equations. In this project we are mainly concerned with the symbolic treatment of ODEs and PDEs, i.e. reductions of differential problems to algebraic ones and algebraic methods for these derived problems. This idea of an algebraic approach to differential equations has a long history. In the 19th century S. Lie initiated the investigation of transformations, which leave a given differential equation invariant. Such transformations are commonly known as Lie symmetries. They form a group, a so-called Lie group. The basic idea here is to find a group of symmetries of the differential equations and then use this group to reduce the order or the number of variables appearing in the equation. At the beginning of the 20th century Riquier and Janet introduced the concept of involutivity for systems of algebraic differential equations. Their theory leads to canonical systems of generators for differential ideals, and the algorithm for generating such canonical Janet bases is strikingly similar to the method of Gröbner bases for generating canonical systems for algebraic ideals devoloped by Buchberger. Another approach to symbolic solution of differential equations has been initiated by J. Liouville for the study of formal integration, and Liouville`s method has been extended to an algorithm for solving linear differential equations. There are several advantages of an algebraic analysis of differential equations: sometimes we are actually able to go the whole way and find a symbolic solution to the given DE; if we can derive symmetries of a DE, then this helps a lot in verifying numerical schemes for approximation of solutions; we can decide whether a system of ODEs or PDEs admits a solution; if there are solutions, then we can derive differential systems in triangular form such that the solutions of the original system are the (non-singular) solutions of the output system; we can get a complete overview of the algebraic relations satisfied by the solutions of a given system of differential algebraic equations (DAEs). An effective treatment of these problems depends crucially on algorithms in differential elimination theory, i.e. the theory of differential ideals in a ring of differential polynomials. The algebraic theory of elimination is well developed, and we can answer the main questions such as consistency of ideals, equality of ideals, the membership problem, the radical membership problem, the arithmetic of ideals, etc. constructively. The methods for solving these problems are resultants, Gröbner bases, and algebraic involutive bases. For differential ideals there are still many open problems. For instance, the membership problem or the ideal inclusion problems for finitely generated differential ideals are still not solved. The analysis and improvement of existing algorithms as well as the development of new algorithmic methods in differential elimination theory is the major goal of the project. Additionally, we will also consider the important application area of control theory.