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Cylindrical algebraic decomposition is one of the most important tools for computing with semi-algebraic sets, while triangular decomposition is among the most important approaches for manipulating constructible sets. In this paper, for an arbitrary finite set <i>F</i> ⊂ [<i>y<sub>1</sub></i>,...,<i>y<sub>n</sub></i>] we apply comprehensive triangular… (More)

We introduce the concept of comprehensive triangular decomposition (CTD) for a parametric polynomial system F with coefficients in a field. In broad words, this is a finite partition of the the parameter space into regions, so that within each region the " geometry " (number of irreducible components together with their dimensions and degrees) of the… (More)

We present an algorithm RealRootIsolate for isolating the real roots of a system of multivariate polynomials given by a zerodimensional squarefree regular chain. The output of the algorithm is guaranteed in the sense that all real roots are obtained and are described by boxes of arbitrary precision. Real roots are encoded with a hybrid representation,… (More)

Regular chains and triangular decompositions are fundamental and well-developed tools for describing the complex solutions of polynomial systems. This paper proposes adaptations of these tools focusing on solutions of the real analogue: semi-algebraic systems.
We show that any such system can be decomposed into finitely many <i>regular semi-algebraic… (More)

We discuss algorithmic advances which have extended the pioneer work of Wu on triangular decompositions. We start with an overview of the key ideas which have led to either better implementation techniques or a better understanding of the underlying theory. We then present new techniques that we regard as essential to the recent success and for future… (More)

This article is a continuation of our earlier work [3], which introduced triangular decompositions of semi-algebraic systems and algorithms for computing them. Our new contributions include theoretical results based on which we obtain practical improvements for these decomposition algorithms.
We exhibit new results on the theory of <i>border… (More)

A CAD of R n is a partition of R n such that each cell in the partition is a connected semi-algebraic subset of R n and all the cells are cylindrically arranged. Two subsets A and B of R n are called cylindrically arranged if for any 1 ≤ k < n, the projections of A and B on R k are either equal or disjoint.

We propose new algorithms for computing triangular decompositions of polynomial systems incrementally. With respect to previous works, our improvements are based on a <i>weakened</i> notion of a polynomial GCD modulo a regular chain, which permits to greatly simplify and optimize the sub-algorithms. Extracting common work from similar expensive computations… (More)

A new algorithm to compute cylindrical algebraic decompo-sitions (CADs) is presented, building on two recent advances. Firstly, the output is truth table invariant (a TTICAD) meaning given formulae have constant truth value on each cell of the decomposition. Secondly, the computation uses regular chains theory to first build a cylindrical decomposition of… (More)

- Changbo Chen, Marc Moreno Maza, Rob Corless
- 2011