On irreducible components of real exponential hypersurfaces

Abstract

Fix any algebraic extension $\mathbb{K}$ of the field $\mathbb{Q}$ of rationals. In this article we study exponential sets $V\subset {\mathbb{R}}^n$. 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 ${\mathbb{R}}^n$ form a Noethrian topology on ${\mathbb{R}}^n$, which we will call Zariski topology. Let $P\in \mathbb{K}[X_1,\dots ,X_n,U_1,\dots ,U_n]$ be a polynomial such that $$V=\{\mathbf{x}=(x_1,\dots ,x_n)\in {\mathbb{R}}^n|P(\mathbf{x},e^{x_1},\dots ,e^{x_n})=0\}.$$ 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 $\mathrm{Zer}(\mathrm{P})$ 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 $U_2,\dots ,U_n$) stronger statements are shown which are independent of Schanuel's conjecture.

arXiv commentSome minor changes. Final version, to appear in Arnold Mathematical Journal