On Irreducible Components of Real Exponential Hypersurfaces

Cordian Riener, Nicolai Vorobjov*

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

Abstract

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 overK. 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 (Formula presented.) 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.

Original languageEnglish
Pages (from-to)423-443
Number of pages21
JournalArnold Mathematical Journal
Volume3
Issue number3
DOIs
Publication statusPublished - 1 Sep 2017
MoE publication typeA1 Journal article-refereed

Keywords

  • Exponential-algebraic set
  • Irreducible components
  • Schanuel’s conjecture

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