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 over K. We study zero sets in Rn of exponential polynomials over K, which we call exponentialalgebraic sets. Complements of all exponentialalgebraic sets in Rn form a Zariskitype topology on Rn. Let P∈K[X1,…,Xn,U1,…,Un] be a polynomial and denote
V:={(x1,…,xn)∈RnP(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 exponentialalgebraic 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.
V:={(x1,…,xn)∈RnP(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 exponentialalgebraic 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 language  English 

Pages (fromto)  423443 
Number of pages  21 
Journal  Arnold Mathematical Journal 
Volume  3 
Early online date  9 Aug 2017 
DOIs  
Publication status  Published  10 Sept 2017 
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Nicolai Vorobjov
Person: Research & Teaching