In
mathematics, the constant problem is the problem of
deciding whether a given expression is equal to
zero
0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usu ...
.
The problem
This problem is also referred to as the identity problem or the method of zero estimates. It has no formal statement as such but refers to a general problem prevalent in
transcendental number theory
Transcendental number theory is a branch of number theory that investigates transcendental numbers (numbers that are not solutions of any polynomial equation with rational coefficients), in both qualitative and quantitative ways.
Transcendence
...
. Often proofs in transcendence theory are
proofs by contradiction. Specifically, they use some
auxiliary function to create an
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
''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
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
s,
limits,
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An ex ...
s, other
functions, and
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if ...
s 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 ''x''
1, ..., ''x''
''n'' are
real number
In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
s, 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
''Mathematics of Computation'' is a bimonthly mathematics journal focused on computational mathematics. It was established in 1943 as ''Mathematical Tables and other Aids to Computation'', obtaining its current name in 1960. Articles older than f ...
, volume=50 , issue=20 , date=January 1988 , pages=275–281 , url=http://www.davidhbailey.com/dhbpapers/const.pdf , doi=10.1090/S0025-5718-1988-0917835-1, doi-access=free for deciding whether there are integers ''a''
1, ..., ''a''
''n'' such that
:
If the expression we are interested in contains an oscillating function, such as the
sine
In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is opp ...
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 An integer relation between a set of real numbers ''x''1, ''x''2, ..., ''x'n'' is a set of integers ''a''1, ''a''2, ..., ''a'n'', not all 0, such that
:a_1x_1 + a_2x_2 + \cdots + a_nx_n = 0.\,
An integer relation algorithm is an algorithm fo ...
References
Analytic number theory
Unsolved problems in mathematics