Cylindrical algebraic decomposition I: the basic algorithm
SIAM Journal on Computing
An improved projection operation for cylindrical algebraic decomposition of three-dimensional space
Journal of Symbolic Computation
An adjacency algorithm for cylindrical algebraic decompositions of three-dimenslonal space
Journal of Symbolic Computation
A cluster-based cylindrical algebraic decomposition algorithm
Journal of Symbolic Computation
Quantifier elimination: Optimal solution for two classical examples
Journal of Symbolic Computation
Towards mechanical solution of the Kahan Ellipse Problem 1
EUROCAL '83 Proceedings of the European Computer Algebra Conference on Computer Algebra
A solution to Kahan's problem (SIGSAM problem no. 9)
ACM SIGSAM Bulletin
Problem #9: an ellipse problem
ACM SIGSAM Bulletin
A cluster-based cylindrical algebraic decomposition algorithm
Journal of Symbolic Computation
A bibliography of quantifier elimination for real closed fields
Journal of Symbolic Computation
Quantifier elimination and the sign variation method for real root isolation
ISSAC '89 Proceedings of the ACM-SIGSAM 1989 international symposium on Symbolic and algebraic computation
A parallel implementation of the cylindrical algebraic decomposition algorithm
ISSAC '89 Proceedings of the ACM-SIGSAM 1989 international symposium on Symbolic and algebraic computation
Automated reasoning in geometries using the characteristic set method and Gröbner basis method
ISSAC '90 Proceedings of the international symposium on Symbolic and algebraic computation
An improvement of the projection operator in cylindrical algebraic decomposition
ISSAC '90 Proceedings of the international symposium on Symbolic and algebraic computation
Partial Cylindrical Algebraic Decomposition for quantifier elimination
Journal of Symbolic Computation
An algorithm for solving parametric linear systems
Journal of Symbolic Computation
ISSAC '92 Papers from the international symposium on Symbolic and algebraic computation
On order-invariance of a binomial over a nullifying cell
ISSAC '03 Proceedings of the 2003 international symposium on Symbolic and algebraic computation
Computation of equilibriain noncooperative games
Computers & Mathematics with Applications
Visually Dynamic Presentation of Proofs in Plane Geometry
Journal of Automated Reasoning
Quantifier elimination for quartics
AISC'06 Proceedings of the 8th international conference on Artificial Intelligence and Symbolic Computation
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We give solutions to two problems of elementary algebra and geometry: (1) find conditlons on real numbers p, q, and r; so that the polynomial function f(x) = x^4 + px^2 + q x+ r is nonnegative for all real x and (2) find conditions on real numbers a, b, and c so that the ellipse (x-c)^2q^2+y^2b^2-1=0 lies inside the unit circle y^2 + x^2 - 1 = O. Our solutions are obtained by following the basic outline of the method of quantifier elimination by cylindrical algebraic decomposition (Collins, 1975), but we have developed, and have been considerably aided by, modified vcrsions of certain of its steps. We have found three equally simple but not obviously equivalent solutions for the first problem, illustrating the difficulty of obtaining unique ''simplest'' solutions to quantifier eliminetion problems of elementary algebra and geometry.