Elimination Methods in Polynomial Computer Algebra

The subject of this book is connected with a new direction in mathematics, which has been actively developed over the last few years, namely the field of polynomial computer algebra, which lies at the intersection point of algebra, mathematical analysis and programming. There were several incentives to write the book. First of all, there has lately been a considerable interest in applied nonlinear problems characterized by multiple sta tionary states. Practical needs have then in their turn led to the appearance of new theoretical results in the analysis of systems of nonlinear algebraic equations. And finally, the introduction of various computer packages for analytic manipulations has made it possible to use complicated elimination-theoretical algorithms in prac tical research. The structure of the book is accordingly represented by three main parts: Mathematical results driven to constructive algorithms, computer algebra realizations of these algorithms, and applications. Nonlinear systems of algebraic equations arise in diverse fields of science. In particular, for processes described by systems of differential equations with a poly nomial right hand side one is faced with the problem of determining the number (and location) of the stationary states in certain sets.

Klappentext

This book presents a modified method, based on multidimensional residue theory, for the elimination of unknowns from a system of nonlinear algebraic equations. An algorithm is given for constructing the resultant of the system, and a computer implementation making use of formula manipulation software is carried out. Programmes in MAPLE are available. The algorithms and programmes are then applied to questions from the theory of chemical kinetics, such as the search for all stationary solutions of kinetic equations and the construction of kinetic polynomials. The subject of this book is closely connected with a wide range of current problems in the analysis of nonlinear systems. br/ emAudience:/em This volume will be of interest to graduate students and researchers whose work involves multidimensional theory of residues, mathematical kinetics, computer algebra, and symbolic computation.

Inhalt

1. Basic Mathematical Facts.- 1. The logarithmic residue.- 2. The Newton recursion formulas.- 3. Localization theorems for the real zeros of a polynomial.- 4. The local residue (of Grothendieck).- 5. The multidimensional logarithmic residue.- 6. The classical scheme for elimination of unknowns.- 2. A Modified Elimination Method.- 7. A generalized transformation formula for local residues.- 8. A modified elimination method.- 9. A formula for the logarithmic derivative of the resultant.- 10. Multidimensional analogues of the Newton formulas.- 11. Elimination of unknowns in different variables. Real roots.- 3. Applications in Mathematical Kinetics.- 12. Short schemes.- 13. The search for all stationary solutions.- 14. The kinetic polynomial. Single-route mechanisms.- 15. Construction of the kinetic polynomial in the general case.- 4. Computer Realizations.- 16. Analytic manipulations on the computer.- 17. Basic problems in computer algebra of polynomials.- 18. Realization of the elimination method.- 19. The construction of the resultant.- List of applications.