Monodromy and irreducibility criteria with algorithmic applications in zero characteristic

A. L. Chistov

Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part VII, Tome 292 (2002), pp. 130-152

Résumé

Consider a projective algebraic variety $V$ which is the set of all common zeroes of homogeneous polynomials of degrees less than $d$ in $n+1$ variables in zero-characteristic. We suggest an algorithm to decide whether two (or more) given points of $V$ belong to the same irreducible component of $V$. Besides that we show how to construct for each $s<n$ an $(s+1)$-dimensional plane in the projective space such that the intersection of every irreducible component of dimension $n-s$ of $V$ with the constructed plane is transversal and is an irreducible curve. These algorithms are deterministic and polynomial in $d^n$ and the size of input.