2016/09/20 by Riener, Cordian, Vorobjov, Nicolai
#11J81 #14P15 #Algebraic Geometry (math.AG) #FOS: Mathematics #Logic (math.LO)
paper · doi:10.48550/arxiv.1609.06025
Fix any algebraic extension \mathbb K of the field \mathbb Q of rationals. In this article we study exponential sets V⊂ \mathbb Rn. Such sets are described by the vanishing of so called exponential polynomials, i.e., polynomials with coefficients from \mathbb K, in n variables, and in n exponential functions. The complements of all exponential sets in \mathbb Rn form a Noethrian topology on \mathbb Rn, which we will call Zariski topology. Let P ∈ \mathbb K[X1, … ,Xn,U1, … ,Un] be a polynomial such that V=\ x=(x1, … , xn) ∈ \mathbb Rn| P(x, ex1, … ,exn)=0 \. The main result of this paper states that, under Schanuel's conjecture over the reals, an exponential set V of codimension 1, for which the real algebraic set \rm Zer(P) is irreducible over \mathbb K, either is irreducible (with respect to the Zariski topology) or every of its irreducible components of codimension 1 is a rational hyperplane through the origin. The family of all possible hyperplanes is determined by monomials of P. In the case of a single exponential (i.e., when P is independent of U2, … , Un) stronger statements are shown which are independent of Schanuel's conjecture.