Explicit estimates for polynomial systems defining irreducible smooth complete intersections

Abstract

This paper deals with properties of the algebraic variety defined as the set of zeros of a “typical” sequence of polynomials. We consider various types of “nice” varieties: set-theoretic and ideal-theoretic complete intersections, absolutely irreducible ones, and nonsingular ones. For these types, we present a nonzero “genericity” polynomial of explicitly bounded degree in the coefficients of the sequence that vanishes if its variety is not of the type. Here, the number of polynomials and their degrees are fixed. Over finite fields, this yields bounds on the number of such sequences. We also show that most sequences (of at least two polynomials) define a degenerate variety, namely an absolutely irreducible nonsingular hypersurface in some linear projective subspace.