From Schanuel’s Conjecture to Shapiro’s Conjecture

Abstract

In this paper we prove Shapiro's 1958 Conjecture on exponential polynomials, assuming Schanuel's Conjecture.

Hyperbolic polynomials and linear-type generating functions

“…Without striving for a completeness, we mention only a few here. Shapiro's 1958 conjecture on the zero distribution of the members of the ring or exponential polynomials motivated D'Aquino, Macintyre and Terzo to study these objects in an algebraic setting in the paper [1]), where they attribute the origins of Shapiro's conjecture to complex analytic considerations. Exponential polynomials are also central objects in the study of decomposition of integers into sums of powers of integers (such as Vinogradov's Three primes theorem) and in the methods used in arriving at such theorems (such as the Hardy-Littlewood circle method).…”

Generic solutions of equations with iterated exponentials

“…In this paper we consider the next natural cases of exponential polynomials with two and three iterations of exponentations, and we obtain an analogous result to that of Marker. Comparing the complex exponential field and Zilber's fields has been one of the main motivation in the following recent papers [3], [5], [8], [13].…”

Comparing and Zilber’s Exponential Fields: Zero Sets of Exponential Polynomials

“…Reduction 2. This involves the very first step in our proof of Shapiro's Conjecture from Schanuel's Conjecture (see Section 5 of [7]). Here we have the hypothesis that the transcendence degree of X is finite in contrast to our previous paper on Shapiro's Conjecture where we have to prove that the set of common zeros of two exponential polynomials have finite transcendence degree.…”

“…More recently in [7] we derived, in an exponentialalgebraic way, Shapiro's Conjecture from Schanuel's Conjecture, thereby getting Shapiro's Conjecture in B. In the present paper we put the ideas of [7] to work on some problems connected to the Identity Theorem of complex analysis [1]. That fundamental theorem says in particular that if the zero set of an entire function f has an accumulation point then f ≡ 0.…”

“…An early success was a proof in B of the Schanuel's Nullstellensatz [6], proved in C ( [13]) using Nevanlinna theory. More recently in [7] we derived, in an exponentialalgebraic way, Shapiro's Conjecture from Schanuel's Conjecture, thereby getting Shapiro's Conjecture in B. In the present paper we put the ideas of [7] to work on some problems connected to the Identity Theorem of complex analysis [1].…”