Computing the Canonical Representation of Constructible Sets

In the basic paper defining the Gröbner Cover [16] for discussing parametric polynomial systems of equations, we 8 introduced algorithms that have been improved since then. We used our own algorithm buildtree for computing 9 the initial Comprehensive Gröbner System (CGS), needed for the Gröbner Cover, now substituted in the Singular [7] 10 library “grobcov.lib” by themore efficientKapur–Sun–Wang algorithm [11]. The algorithmgrobcov used specially 11 simple locally closed sets, whose union is certified to be also locally closed by Wibmer’s theorem [17] (algorithm 12 LCUnion). 13 The Gröbner Cover is used in [15] for the automatic deduction of geometric theorems. It is also essential for 14 computing geometrical loci and defining a taxonomy of the components of loci in [1], as well as for envelopes. 15 In general in these tasks, the representation of locally closed sets, i.e. difference of varieties, is sufficient. But for 16 more general applications, where Wibmer’s theorem [17] is not applicable, the union of locally closed sets is not 17 always locally closed. This is the reason for reviewing here the canonical representation of constructible sets giving 18 algorithms to compute it, as well as to use the new algorithms inside the library for computing higher dimensional 19