A note on algebraic independence of logarithmic and exponential constants

Abstract

This paper gives a corollary to Schanuel's conjecture that indicates when an exponential or logarithmic constant is transcendental over a given field of constants. The given field is presumed to have been built up by starting with the rationals Q with π adjoined and taking algebraic closure, adjoining values of the exponential function or of some fixed branch of the logarithmic function, and then repeating these two operations a finite number of times.

A differential-equations approach to functional equivalence

“…It should be pointed out that proof of this lies well beyond tbe current state of development of transcendental number theory, but it is at least a plausible and long-standing conjecture; see [12], [l], [7], [3]. However, one can hardly make a similar conjecture concerning extensions via arbitrary solutions of firstorder differential equations (let alone bigher orders) and SO the question is very much open as to what, in practice, one should do about constants at this level of generality.…”

Zero Tests for Constants in Simple Scientific Computation

“…, e x k } contains at least k algebraically independent numbers. This has been used to solve zero problems, related to the problem for the closed form numbers, by Caviness and Prelle [7], by Wilkie and Macintyre [13] and also by Richardson [14].…”

Zero-equivalence in function fields defined by algebraic differential equations

“…This is clearly necessary anyway, since an algorithm to decide zero-equivalence must in particular be applicable to constant expressions. This constants problem is related to some old unsolved problems in transcendental number theory [1,17,7,9] and looks very difficult. For this reason much of the existing work on zero-equivalence in transcendental function fields has proceeded from the (quite unwarranted) assumption that an oracle exists for the constants.…”

“…We shall henceforth adopt this standpoint here, but the point should be made that because of the greater generality, this involves a somewhat larger assumption in our case. In particular, if the function field is generated by logarithms and exponentials only, constants may be dealt with by assuming the Schanuel conjecture, [17,9]. This may be stated as follows:…”