Computations with parametric equations
ISSAC '91 Proceedings of the 1991 international symposium on Symbolic and algebraic computation
Solving parametric algebraic systems
ISSAC '92 Papers from the international symposium on Symbolic and algebraic computation
Journal of Symbolic Computation
A generalized Euclidean algorithm for computing triangular representations of algebraic varieties
Journal of Symbolic Computation
Algebraic numbers: an example of dynamic evaluation
Journal of Symbolic Computation
Decomposing polynomial systems into simple systems
Journal of Symbolic Computation
On the theories of triangular sets
Journal of Symbolic Computation - Special issue on polynomial elimination—algorithms and applications
Computing triangular systems and regular systems
Journal of Symbolic Computation
A new algorithm for discussing Gröbner bases with parameters
Journal of Symbolic Computation
Canonical comprehensive Gröbner bases
Proceedings of the 2002 international symposium on Symbolic and algebraic computation
Factoring into coprimes in essentially linear time
Journal of Algorithms
A simple algorithm to compute comprehensive Gröbner bases using Gröbner bases
Proceedings of the 2006 international symposium on Symbolic and algebraic computation
Proceedings of the twenty-first international symposium on Symbolic and algebraic computation
On the representation of constructible sets
ACM Communications in Computer Algebra
The ConstructibleSetTools and ParametricSystemTools modules of the RegularChains library in Maple
ACM Communications in Computer Algebra
Computing cylindrical algebraic decomposition via triangular decomposition
Proceedings of the 2009 international symposium on Symbolic and algebraic computation
Computations modulo regular chains
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
Gröbner bases for polynomial systems with parameters
Journal of Symbolic Computation
Thomas decomposition of algebraic and differential systems
CASC'10 Proceedings of the 12th international conference on Computer algebra in scientific computing
Some notes upon "When does equal sat(T)?"
AISC'10/MKM'10/Calculemus'10 Proceedings of the 10th ASIC and 9th MKM international conference, and 17th Calculemus conference on Intelligent computer mathematics
Computing with semi-algebraic sets represented by triangular decomposition
Proceedings of the 36th international symposium on Symbolic and algebraic computation
Algorithms for computing triangular decompositions of polynomial systems
Proceedings of the 36th international symposium on Symbolic and algebraic computation
Computing comprehensive Gröbner systems and comprehensive Gröbner bases simultaneously
Proceedings of the 36th international symposium on Symbolic and algebraic computation
On the regularity property of differential polynomials modulo regular differential chains
CASC'11 Proceedings of the 13th international conference on Computer algebra in scientific computing
Semi-algebraic description of the equilibria of dynamical systems
CASC'11 Proceedings of the 13th international conference on Computer algebra in scientific computing
Journal of Symbolic Computation
Algorithms for computing triangular decomposition of polynomial systems
Journal of Symbolic Computation
Algorithmic Thomas decomposition of algebraic and differential systems
Journal of Symbolic Computation
Triangular decomposition of semi-algebraic systems
Journal of Symbolic Computation
Journal of Symbolic Computation
An efficient method for computing comprehensive Gröbner bases
Journal of Symbolic Computation
Computer algebra methods in the study of nonlinear differential systems
Computational Mathematics and Mathematical Physics
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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 algebraic variety of the specialized system F(u) is the same for all values u of the parameters. We propose an algorithm for computing the CTD of F. It relies on a procedure for solving the following set theoretical instance of the coprime factorization problem. Given a family of constructible sets A1,..., As, compute a family B1,..., Bt of pairwise disjoint constructible sets, such that for all 1 ≤ i ≤ s the set Ai writes as a union of some of the B1,...,Bt. We report on an implementation of our algorithm computing CTDs, based on the RegularChains library in MAPLE. We provide comparative benchmarks with MAPLE implementations of related methods for solving parametric polynomial systems. Our results illustrate the good performances of our CTD code.