Constant problem

In mathematics, the constant problem is the problem of deciding if a given expression is equal to zero.

The problem

This problem is also referred to as the identity problem^[1] or the method of zero estimates. It has no formal statement as such but refers to a general problem prevalent in transcendental number theory. Often proofs in transcendence theory are proofs by contradiction, specifically they use some auxiliary function to create an integer n ≥ 0 which is shown to satisfy n < 1. Clearly this means that n must have the value zero, and so a contradiction arises if one can show that in fact n is not zero.

In many transcendence proofs, proving that n ≠ 0 is very difficult, and hence a lot of work has been done to develop methods that can be used to prove the non-vanishing of certain expressions. The sheer generality of the problem is what makes it difficult to prove general results or come up with general methods for attacking it. The number n that arises may involve integrals, limits, polynomials, other functions, and determinants of matrices.

Results

In certain cases algorithms or other methods exist for proving that a given expression is non-zero, or of showing that the problem is undecidable. For example, if x₁, ..., x_n are real numbers then there is an algorithm^[2] for deciding if there are integers a₁, ..., a_n such that

$|a_1 x_1+{}\cdots{}+a_n x_n|=0.\,$

If the expression we are interested in contains a function which oscillates, such as the sine or cosine function, then it has been shown that the problem is undecidable, a result known as Richardson's theorem. In general, methods specific to the expression being studied are required to prove that it cannot be zero.

See also

• Integer relation algorithm

References

1. Richardson, Daniel (1968). "Some Unsolvable Problems Involving Elementary Functions of a Real Variable". Journal of Symbolic Logic 33: 514–520. doi:10.2307/2271358.
2. Bailey, David H. (January 1988). "Numerical Results on the Transcendence of Constants Involving π, e, and Euler's Constant" (PDF). Mathematics of Computation 50 (20): 275–281. doi:10.1090/S0025-5718-1988-0917835-1.