On irreducible components of real exponential hypersurfaces

TL;DR

This paper investigates the structure of real exponential hypersurfaces, showing that under Schanuel's conjecture, codimension 1 exponential sets are either irreducible or composed of rational hyperplanes, with stronger results for single exponential cases.

Contribution

It provides a classification of irreducible components of exponential hypersurfaces under Schanuel's conjecture, linking algebraic irreducibility to geometric hyperplanes.

Findings

Under Schanuel's conjecture, exponential hypersurfaces of codimension 1 are either irreducible or unions of rational hyperplanes.

In the single exponential case, stronger, conjecture-independent results are established.

The structure of hyperplanes is explicitly determined by monomials of the defining polynomial.

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 $P \in {\mathbb K}[X_1, \ldots ,X_n,U_1, \ldots ,U_n]$ be a polynomial such that $$V=\{ \mathbf{x}=(x_1, \ldots , x_n) \in \mathbb R^n| P(\mathbf{x}, e^{x_1}, \ldots ,e^{x_n})=0 \}.$$ 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 $\rm Zer(P)$ 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 $U_2, \ldots , U_n$) stronger statements are shown which are independent of Schanuel's conjecture.