mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, 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. Adding (or subtracting) 0 to any number leaves that number unchanged; in mathematical terminology, 0 is the additive identity of the integers, rational numbers, real numbers, and compl ...

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. Transcendenc ...

. 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 (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

''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 is the continuous analog of a Summation, sum, which is used to calculate area, areas, volume, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental oper ...

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 2009 ...

polynomial In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...

s, other functions, and

determinant In mathematics, the determinant is a Scalar (mathematics), scalar-valued function (mathematics), function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the ...

s of

matrices Matrix (: matrices or matrixes) or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the ...

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 duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...

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=https://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 opposite th ...

cosine 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 opposite that ...

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.

References

Analytic number theory Undecidable problems

The problem

Results

See also

References