On Irreducible Components of Real Exponential Hypersurfaces

Tutkimustuotos: Lehtiartikkelivertaisarvioitu


  • Cordian Riener
  • Nicolai Vorobjov


  • UiT The Arctic University of Norway
  • University of Bath


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.


JulkaisuArnold Mathematical Journal
TilaJulkaistu - 1 syyskuuta 2017
OKM-julkaisutyyppiA1 Julkaistu artikkeli, soviteltu

ID: 16606608