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Elimination methods. (English) Zbl 0964.13014
The task of solving a system of polynomial equations, or more generally of extracting information from it, arises in a variety of situations, ranging from fields of pure mathematics like algebraic geometry to applied mathematics and to technical applications like robotics. Elimination is a technique for addressing these problems which has a long history. Born from the geometric idea of considering the projection of a given object or, equivalently, the intersection of an ideal in a polynomial ring with a subring involving fewer variables, it has been an important constructive tool for a long time. After loosing importance when more elegant abstract arguments appeared in commutative ring theory, a renaissance of elimination methods followed the progress of computer technology, since it is nowadays feasible to perform the rather complex computations algorithmically on a computer.
The book “Elimination methods” by D. Wang under review presents different elimination algorithms that are presently in use. After briefly introducing basic concepts and properties of multivariate polynomials, the author focuses on algorithms which decompose arbitrary polynomial systems into triangular systems, that is, into systems of polynomials \(T_1,\dots,T_r\) in which the polynomial \(T_i\) only involves the first \(p_i\) variables of the base ring \((p_1 < \ldots < p_r)\). Before proceeding to some applications of elimination in the last chapters, other better-known algorithms based on Gröbner bases and on resultant techniques are explained, but not in as much detail as in the discussion of triangular systems.
The discussion of each method in this book also contains a formal description of the algorithm, which allows a rather direct implementation, and some examples illustrating it; complexity questions and practical implementation are not covered in this book.

13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
13-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to commutative algebra
68-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to computer science
14Q99 Computational aspects in algebraic geometry
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
68W30 Symbolic computation and algebraic computation