Numerically intersecting algebraic varieties via witness sets (Q2434875)

The article is devoted to describe an algorithm for computing the intersection \(A\cap B\) of two irreducible sets \(A\subset\mathbb{C}^N\) and \(B\subset\mathbb{C}^N\) given by witness sets. A witness set for an irreducible set \(A\subset\mathbb{C}^N\) is a triple \(W:=\{F,\mathcal{L},\mathcal{L}\cap A\}\), where \(F:\mathbb{C}^N\to\mathbb{C}^N\) is a polynomial mapping having \(A\) as an irreducible component of \(\mathcal{V}(F):=\{x\in\mathbb{C}^N:F(x)=0\}\), \(\mathcal{L}\subset\mathbb{C}^N\) is a generic linear space with \(\dim\mathcal{L}=N-\dim A\) and \(\mathcal{L}\cap A\) is a finite set of \(\deg A\) points. Algorithms for computing the intersection \(A\cap B\) from witness sets for \(A\) and \(B\) were given in [\textit{A. Sommese} et al., SIAM J. Numer. Anal. 42, No. 4, 1552--1571 (2004; Zbl 1108.13308), J. Complexity 16, No. 3, 572--602 (2005; Zbl 1108.13309)]. Such algorithms, called \textit{diagonal homotopy} methods, compute a collection of witness \textit{supersets}, one for each irreducible component of \(A\cap B\). Nevertheless, these algorithms fail to obtain a collection of witness sets in certain cases. In the paper under review, the algorithms of the above cited papers are modified in order to obtain a witness set for each irreducible component of \(A\cap B\). A problem with the previous algorithms is that they are not able to properly deal with points in a witness superset for an irreducible component of \(A\cap B\) which are not smooth points of the solution system \(\mathcal{V}(F)\) for \(A\) and \(B\). For this purpose, the authors rely on the theory of \textit{isosingular sets} [\textit{J. Hauenstein} and \textit{C. Wampler}, Found. Comput. Math. 13, No. 3, 371--403 (2013; Zbl 1276.65029)]. An isosingular set of a given algebraic set is an irreducible set which is constructed by taking the Zariski closure of the set of points with a common singularity structure. This singularity structure is represented by a sequence of nonnegative integers called the \textit{deflation sequence} of \(A\), where a \textit{deflation} is a method for regularizing a nonreduced solution set \(\mathcal{V}(F)\). Combining a deflation method of Hauenstein and Wampler [loc. cit.] with simplified versions of the \textit{diagonal homotopy} methods mentioned above, the authors are able to extract a witness set for each irreducible component of \(A\cap B\) from a corresponding witness superset. The paper ends by illustrating concepts and algorithms on a simple example and a more complicated example taken from kinematics.