On the number of zeros of general exponential polynomials

On polynomials and exponential polynomials in several complex variables

“…In the case of exponential polynomials, our results can be seen as partial generalizations of Tijdeman's well-known estimates for a single complex variable [9], and it appears that in fact they suffice for several applications to transcendence theory. Recently Waldschmidt [13] has settled an old problem of Weil and Serre on characters by combining our zero-estimates with an ingenious idea of his own.…”

“…The well-known estimates [9] of Tijdeman for a single complex variable are also essentially best possible, and it is interesting to compare them with our Theorem2 for d=l. In this case the integers rn~,...,m, satisfy ml<l, m 2 ..... rrtn=0 , whence z=m-1 or m/n.…”

A Conjecture by Leon Ehrenpreis About Zeroes of Exponential Polynomials

“…Montgomery in a colloquium in Number Theory (Bolyai Janos ed. ), see [57,58]. Conjecture 1.5 (Montgomery-Shapiro conjecture) Let f, g be two exponential polynomials that have an infinite number of common zeroes.…”

Fields of large transcendence degree generated by values of elliptic functions

“…Actually all our difficulties arise from one basic fact: there is no known elliptic analogue of Tijdeman's results [24] for small values of exponential polynomials. Even for the underlying zero estimates [23] the only satisfactory substitute at the moment comes from the general work of the present authors [17] on group varieties.…”