A Zero Test for Exp Log Constants
Daniel Richardson
Computer Science, Bath University, U.K.
The Exp Log constants are the smallest collection of expressions which
contains the rational numbers and is closed under addition,
subtraction, multiplication, division, ExP, Log, and radicals. These
expressions represent real or complex numbers, with some ambiguity
because of the presence of notations for multivalued functions. The
Zero problem for these constants is to decide whether or not the value
of an expression E is actually zero, given E and a partial evaluation
which is sufficient to remove ambiguitity about the value. A zero
test is stated which attempts to solve this problem. The test is
correct in that whenever it terminates it gives the correct value. If
the Schanuel conjecture is true, the test will always eventually
terminate. So the test is complete if the Schanuel conjecture is
true. The algebraic machinery needed for the test is the ability to
decide whether or not an algebraic function of several variables,
given in terms of field operations and radicals, is identically zero.