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{{Short description|Problem of deciding whether an expression equals zero}}
{{confusing|date=August 2016}}
In [[mathematics]], the '''constant problem''' is the problem of [[Decidability (logic)|deciding]] whether a given expression is equal to [[zero]].

==The problem==
This problem is also referred to as the '''identity problem'''<ref>{{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 }}</ref> 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 [[Reductio ad absurdum|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 [[integral]]s, [[Limit (mathematics)|limits]], [[polynomial]]s, other [[Function (mathematics)|functions]], and [[determinant]]s of [[Matrix (mathematics)|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 problem|undecidable]]. For example, if ''x''1, ..., ''x''''n'' are [[real number]]s, then there is an algorithm<ref>{{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 }}</ref> for deciding whether there are integers ''a''1, ..., ''a''''n'' such that

: <math>a_1 x_1 + \cdots + a_n x_n = 0\,.</math>

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.

==See also==
* [[Integer relation algorithm]]

==References==
<references />

[[Category:Analytic number theory]]
[[Category:Undecidable problems]]

Constant problem - Revision history

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