Quantifier elimination for real closed fields by cylindrical algebraic decomposition

Tarski in 1948, (Tarski 1951) published a quantifier elimination method for the elementary theory of real closed fields (which he had discovered in 1930). As noted by Tarski, any quantifier elimination method for this theory also provides a decision method, which enables one to decide whether any sentence of the theory is true or false. Since many important and difficult mathematical problems can be expressed in this theory, any computationally feasible quantifier elimination algorithm would be of utmost significance.

[1]  A. Tarski.  A Decision Method for Elementary Algebra and Geometry , 2023 .

[2]  Joseph F. Traub,et al.  On Euclid's Algorithm and the Theory of Subresultants , 1971, JACM.

[3]  G. E. Collins.  Computer Algebra of Polynomials and Rational Functions , 1973 .

[4]  W. S. Brown,et al.  On Euclid's Algorithm and the Computation of Polynomial Greatest Common Divisors , 1971, JACM.

[5]  George E. Collins,et al.  Subresultants and Reduced Polynomial Remainder Sequences , 1967, JACM.

[6]  W. Rogosinski,et al.  The Geometry of the Zeros of a Polynomial in a Complex Variable , 1950, The Mathematical Gazette.

[7]  George E. Collins,et al.  The Calculation of Multivariate Polynomial Resultants , 1971, JACM.

[8]  Paul J. Cohen,et al.  Decision procedures for real and p‐adic fields , 1969 .

[9]  Donald Ervin Knuth,et al.  The Art of Computer Programming , 1968 .

[10]  David R. Musser,et al.  Multivariate Polynomial Factorization , 1975, JACM.

[11]  George E. Collins,et al.  Quantifier elimination for real closed fields by cylindrical algebraic decomposition--preliminary report , 1974, SIGS.

[12]  A. Seidenberg.  A NEW DECISION METHOD FOR ELEMENTARY ALGEBRA , 1954 .

[13]  M. Fischer,et al.  SUPER-EXPONENTIAL COMPLEXITY OF PRESBURGER ARITHMETIC , 1974 .

[14]  M. Mignotte.  An inequality about factors of polynomials , 1974 .

[15]  Cyrenus Matthew Rubald.  Algorithms for polynomials over a real algebraic number field. , 1973 .

[16]  Ellis Horowitz,et al.  The minimum root separation of a polynomial , 1974 .

[17]  Jeanne Ferrante,et al.  A Decision Procedure for the First Order Theory of Real Addition with Order , 1975, SIAM J. Comput..