On irreducible components of real exponential hypersurfaces

Fix any real algebraic extension K of the field Q of rationals. Polynomials with coefficients from K in n variables and in n exponential functions are called exponential polynomials over K. We study zero sets in Rn of exponential polynomials over K, which we call exponential-algebraic sets. Complements of all exponential-algebraic sets in Rn form a Zariski-type topology on Rn. Let P∈K[X1,…,Xn,U1,…,Un] be a polynomial and denote

V:={(x1,…,xn)∈Rn|P(x1,…,xn,,ex1,…,exn)=0}.
The main result of this paper states that, if the real zero set of a polynomial P is irreducible over K and the exponential-algebraic set V has codimension 1, then, under Schanuel’s conjecture over the reals, either V is irreducible (with respect to the Zariski topology) or each 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.