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
On mechanical quantifier elimination for elementary algebra and geometry
Journal of Symbolic Computation
Simple CAD construction and its applications
Journal of Symbolic 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
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The improved projection operation for cylindrical algebraic decomposition (CAD) described in [10] requires for its validity the crucial concept of order-invariance. A real polynomial f(x1, ..., xr) is said to be order-invariant in a subset c of Rr if the order (of vanishing) of f at the point p is constant as p varies throughout c. The application of the improved projection which is perhaps simplest conceptually is in the construction of a CAD for a set of polynomials which is well-oriented in a certain sense. Given a well-oriented set A of r-variate integral polynomials algorithm CADW [10] uses the improved projection to construct a CAD of Rr which is order-invariant for each polynomial in A. A drawback of CADW is that it halts in failure, reporting that A is not well-oriented, when presented with a non-well-oriented set A as input. Such failure of CADW is potentially serious, because it forces the user to fall back on less efficient projection operators (such as the Collins-Hong projection) for CAD. The present paper describes an efficient method for avoiding the failure of CADW in certain special non-well-oriented cases. The method is based upon a sufficient criterion for the order of a polynomial h(x1, ..., xr) of the special form h = a(x1, ..., xr-1)xrk + b(x1, ldots, xr-1) (which we shall call a binomial in xr) to be 1 throughout the entire cylinder over a nullifying point p in Rr-1 for h (that is, a point p for which h(p, xr) = 0 identically). Following a review of CADW, the sufficient criterion referred to is motivated and proved. The algorithmic application of the criterion is carefully described and validated, and an example discussed.