Monodromy and irreducibility criteria with algorithmic applications in zero characteristic

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.