constant problem
{{Short description|Problem of deciding whether an expression equals zero}}
{{confusing|date=August 2016}}
In mathematics, the constant problem is the problem of deciding whether a given expression is equal to zero.
The problem
This problem is also referred to as the identity problem{{Cite journal | first=Daniel | last=Richardson | title=Some Unsolvable Problems Involving Elementary Functions of a Real Variable | journal=Journal of Symbolic Logic | volume=33 | year=1968 | pages=514–520 | doi=10.2307/2271358| jstor=2271358 }} 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, ..., xn are real numbers, then there is an algorithm{{Cite journal | first=David H. | last=Bailey | title=Numerical Results on the Transcendence of Constants Involving π, e, and Euler's Constant | journal=Mathematics of Computation | volume=50 | issue=20 | date=January 1988 | pages=275–281 | url=https://www.davidhbailey.com/dhbpapers/const.pdf | doi=10.1090/S0025-5718-1988-0917835-1| doi-access=free }} for deciding whether there are integers a1, ..., an such that
:
If the expression we are interested in contains an oscillating function, 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.