Explicit estimates for polynomial systems defining irreducible smooth complete intersections

TL;DR

This paper provides explicit degree bounds for obstruction polynomials that determine when polynomial systems define certain well-behaved algebraic varieties, with applications to counting such systems over finite fields.

Contribution

It introduces explicit obstruction polynomials with bounded degree that characterize various desirable properties of polynomial systems defining algebraic varieties.

Findings

Explicit obstruction polynomials vanish when the variety is not of the desired type.

Bounds on the number of polynomial sequences over finite fields with specific properties.

Most sequences of at least two polynomials define a degenerate, irreducible hypersurface.

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 "obstruction" polynomial of explicitly bounded degree in the coefficients of the sequence that vanishes if its variety is not of the type. 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.