Elimination theory

Elimination theory is the classical name for algorithmic methods that eliminate one or more variables from a system of polynomial equations in several variables, reducing the system toward a single equation in one remaining variable so that it can be solved. The field reached an early peak in the work of Francis Macaulay on multivariate resultants, recorded in the 1930 edition of Bartel van der Waerden's textbook Moderne Algebra, but fell out of favor for roughly three decades afterward. Elimination theory was revived once computer algebra systems needed genuinely computational methods rather than purely theoretical ones, and two algorithms developed around 1970 became its modern core: Groebner bases, which give an efficient systematic way to eliminate variables from a polynomial ideal, and cylindrical algebraic decomposition, which partitions space into regions where a system of polynomial inequalities keeps a constant sign. Both remain the practical basis for eliminating variables from polynomial systems in current computer algebra software.