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Quantifier elimination is a concept of simplification used in mathematical logic, model theory, and theoretical computer science.Informally, a quantified statement "such that …" can be viewed as a question "When is there an such that …?", and the statement without quantifiers can be viewed as the answer to that question.
In commutative algebra and algebraic geometry, elimination theory is the classical name for algorithmic approaches to eliminating some variables between polynomials of several variables, in order to solve systems of polynomial equations.
For example, although 2 is more than 1, –2 is less than –1. Also if b were zero then zero times anything is zero and cancelling out would mean dividing by zero in that case which cannot be done. So in fact, while cancelling works, cancelling out correctly will lead us to three sets of solutions, not just one we thought we had. It will also ...
The notes were widely imitated, which made (what is now called) Gaussian elimination a standard lesson in algebra textbooks by the end of the 18th century. Carl Friedrich Gauss in 1810 devised a notation for symmetric elimination that was adopted in the 19th century by professional hand computers to solve the normal equations of least-squares ...
Elimination theory, the theory of the methods to eliminate variables between polynomial equations. Disjunctive syllogism, a rule of inference; Gaussian elimination, a method of solving systems of linear equations; Fourier–Motzkin elimination, an algorithm for reducing systems of linear inequalities
For stating the theorem in terms of commutative algebra, one has to consider a polynomial ring [] = [, …,] over a commutative Noetherian ring R, and a homogeneous ideal I generated by homogeneous polynomials, …,. (In the original proof by Macaulay, k was equal to n, and R was a polynomial ring over the integers, whose indeterminates were all the coefficients of the.