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
Partial Cylindrical Algebraic Decomposition for quantifier elimination
Journal of Symbolic Computation
Factors of iterated resultants and discriminants
Journal of Symbolic Computation
On projection in CAD-based quantifier elimination with equational constraint
ISSAC '99 Proceedings of the 1999 international symposium on Symbolic and algebraic computation
Simple CAD construction and its applications
Journal of Symbolic Computation
On propagation of equational constraints in CAD-based quantifier elimination
Proceedings of the 2001 international symposium on Symbolic and algebraic computation
Improved projection for cylindrical algebraic decomposition
Journal of Symbolic Computation
Hauptvortrag: Quantifier elimination for real closed fields by cylindrical algebraic decomposition
Proceedings of the 2nd GI Conference on Automata Theory and Formal Languages
Topologically reliable display of algebraic curves
SIGGRAPH '83 Proceedings of the 10th annual conference on Computer graphics and interactive techniques
QEPCAD B: a program for computing with semi-algebraic sets using CADs
ACM SIGSAM Bulletin
A poly-algorithmic approach to simplifying elementary functions
ISSAC '04 Proceedings of the 2004 international symposium on Symbolic and algebraic computation
Complexity of the resolution of parametric systems of polynomial equations and inequations
Proceedings of the 2006 international symposium on Symbolic and algebraic computation
Combining logical and algebraic techniques for natural style proving in elementary analysis
Mathematics and Computers in Simulation
On delineability of varieties in CAD-based quantifier elimination with two equational constraints
Proceedings of the 2009 international symposium on Symbolic and algebraic computation
Triangular decomposition of semi-algebraic systems
Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation
Efficient preprocessing methods for quantifier elimination
CASC'06 Proceedings of the 9th international conference on Computer Algebra in Scientific Computing
Quantifier elimination for quartics
AISC'06 Proceedings of the 8th international conference on Artificial Intelligence and Symbolic Computation
Triangular decomposition of semi-algebraic systems
Journal of Symbolic Computation
Cylindrical algebraic decompositions for boolean combinations
Proceedings of the 38th international symposium on International symposium on symbolic and algebraic computation
Optimising problem formulation for cylindrical algebraic decomposition
CICM'13 Proceedings of the 2013 international conference on Intelligent Computer Mathematics
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This paper introduces an improved method for constructing cylindrical algebraic decompositions (CADs) for formulas with two polynomial equations as implied constraints. The fundamental idea is that neither of the varieties of the two polynomials is actually represented by the CAD the method produces, only the variety defined by their common zeros is represented. This allows for a substantially smaller projection factor set, and for a CAD with many fewer cells.In the current theory of CADs, the fundamental object is to decompose n-space into regions in which a polynomial equation is either identically true or identically false. With many polynomials, one seeks a decomposition into regions in which each polynomial equation is identically true or false independently. The results presented here are intended to be the first step in establishing a theory of CADs in which systems of equations are fundamental objects, so that given a system we seek a decomposition into regions in which the system is identically true or false --- which means each equation is no longer considered independently. Quantifier elimination problems of this form (systems of equations with side conditions) are quite common, and this approach has the potential to bring large problems of this type into the scope of what can be solved in practice. The special case of formulas containing two polynomial equations as constraints is an important one, but this work is also intended to be extended in the future to the more general case.