Monodromy and irreducibility criteria with algorithmic applications in characteristic zero (Q2577581)

In this interesting paper, the author deals with several algorithmic questions in the context of algebraic geometry. More precisely, if \(k\) is a field of characteristic zero (for the algorithmic part it is assumed to be finitely generated over the field of rational numbers) with algebraic closure \(\overline k\), and \(V\) is a projective algebraic variety in \(\mathbb{P}^n(\overline k)\) given by homogeneous polynomials over \(k\) of degree less that \(d\), the author provides an algorithm to check whether \(z_1,\dots,z_u\in V\) belong to the same irreducible component of \(V\) over \(k\) (resp. over \(\overline k)\). In addition, the author also provides a method to construct \(n-1\) linear forms, with integer coefficients such that for every \(s\in\{0,\dots,n\}\) and for every \(\overline k\)-irreducible component \(W\) of \(V\), of dimension \(n-s\), the intersection of \(W\) with the constructed plane is transversal and is a projective irreducible (over \(\overline k)\) curve. The running time of the first procedure is polynomial in \(d^n\) and in the input size and, concerning to the second, the coefficient length of \(L_i\) is \({\mathcal O}(n\log d)\).