A Deterministic Polynomial-Time Algorithm for the First Bertini Theorem. I

Abstract

Consider a projective algebraic variety W that is an irreducible component of the set of all common zeros of a family of homogeneous polynomials of degree less than d in n + 1 variables in zero characteristic. Consider a linear system on W given by homogeneous polynomials of degree d'. Under the conditions of the first Bertini theorem for W and this linear system, we show how to construct an irreducible divisor in general position from the statement of this theorem. This algorithm is deterministic and polynomial in (dd′)n and the size of the input. This paper is the second part in a tree-part series. Bibliography: 21 titles.