mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the Hindu–Arabic numeral system (or ...

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 In logic and mathematics, proof by contradiction is a form of proof that establishes the truth or the validity of a proposition, by showing that assuming the proposition to be false leads to a contradiction. Proof by contradiction is also known ...

. Specifically, they use some

auxiliary function Auxiliary may refer to: * A backup site or system In language * Auxiliary language (disambiguation) * Auxiliary verb In military and law enforcement * Auxiliary police * Auxiliaries, civilians or quasi-military personnel who provide support of ...

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 Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...

limits Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...

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 exa ...

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 and ...

s of

matrices Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...

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 measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...

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 fi ...

, 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 :

a_1 x_1 + \cdots + a_n x_n = 0\,.

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 oppo ...

or cosine function, then it has been shown that the problem is undecidable, a result known as

Richardson's theorem In mathematics, Richardson's theorem establishes the undecidability of the equality of real numbers defined by expressions involving integers, , \ln 2, and exponential and sine functions. It was proved in 1968 by mathematician and computer scient ...

. In general, methods specific to the expression being studied are required to prove that it cannot be zero.

References

Analytic number theory Unsolved problems in mathematics

The problem

Results

See also

References