On order-invariance of a binomial over a nullifying cell

Abstract

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.