In
mathematics, an integer-valued polynomial (also known as a numerical polynomial)
is a
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 ...
whose value
is 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 ...
for every integer ''n''. Every polynomial with integer
coefficient
In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves ...
s is integer-valued, but the converse is not true. For example, the polynomial
:
takes on integer values whenever ''t'' is an integer. That is because one of ''t'' and
must be an
even number
In mathematics, parity is the property of an integer of whether it is even or odd. An integer is even if it is a multiple of two, and odd if it is not.. For example, −4, 0, 82 are even because
\begin
-2 \cdot 2 &= -4 \\
0 \cdot 2 &= 0 \\
4 ...
. (The values this polynomial takes are the
triangular number
A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The th triangular number is the number of dots i ...
s.)
Integer-valued polynomials are objects of study in their own right in algebra, and frequently appear in
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classif ...
.
[. See in particular pp. 213–214.]
Classification
The class of integer-valued polynomials was described fully by . Inside the
polynomial ring
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variable ...
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
coefficients, the
subring
In mathematics, a subring of ''R'' is a subset of a ring that is itself a ring when binary operations of addition and multiplication on ''R'' are restricted to the subset, and which shares the same multiplicative identity as ''R''. For those ...
of integer-valued polynomials is a
free abelian group
In mathematics, a free abelian group is an abelian group with a basis. Being an abelian group means that it is a set with an addition operation that is associative, commutative, and invertible. A basis, also called an integral basis, is a su ...
. It has as
basis
Basis may refer to:
Finance and accounting
*Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
* Basis trading, a trading strategy consisting o ...
the polynomials
:
for
, i.e., the
binomial coefficient
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
s. In other words, every integer-valued polynomial can be written as an integer
linear combination of binomial coefficients in exactly one way. The proof is by the method of
discrete Taylor series: binomial coefficients are integer-valued polynomials, and conversely, the discrete difference of an integer series is an integer series, so the discrete Taylor series of an integer series generated by a polynomial has integer coefficients (and is a finite series).
Fixed prime divisors
Integer-valued polynomials may be used effectively to solve questions about fixed divisors of polynomials. For example, the polynomials ''P'' with integer coefficients that always take on even number values are just those such that
is integer valued. Those in turn are the polynomials that may be expressed as a linear combination with even integer coefficients of the binomial coefficients.
In questions of prime number theory, such as
Schinzel's hypothesis H and the
Bateman–Horn conjecture In number theory, the Bateman–Horn conjecture is a statement concerning the frequency of prime numbers among the values of a system of polynomials, named after mathematicians Paul T. Bateman and Roger A. Horn who proposed it in 1962. It provides ...
, it is a matter of basic importance to understand the case when ''P'' has no fixed prime divisor (this has been called ''Bunyakovsky's property'', after
Viktor Bunyakovsky
Viktor Yakovlevich Bunyakovsky (russian: Ви́ктор Я́ковлевич Буняко́вский, uk, Ві́ктор Я́кович Буняко́вський; , Bar, Podolia Governorate, Russian Empire – , St. Petersburg, Russian Empire ...
). By writing ''P'' in terms of the binomial coefficients, we see the highest fixed prime divisor is also the highest prime
common factor
In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers ''x'', ''y'', the greatest common divisor of ''x'' and ''y'' i ...
of the coefficients in such a representation. So Bunyakovsky's property is equivalent to coprime coefficients.
As an example, the pair of polynomials
and
violates this condition at
: for every
the product
:
is divisible by 3, which follows from the representation
:
with respect to the binomial basis, where the highest common factor of the coefficients—hence the highest fixed divisor of
—is 3.
Other rings
Numerical polynomials can be defined over other rings and fields, in which case the integer-valued polynomials above are referred to as classical numerical polynomials.
Applications
The
K-theory
In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geom ...
of
BU(''n'') is numerical (symmetric) polynomials.
The
Hilbert polynomial
In commutative algebra, the Hilbert function, the Hilbert polynomial, and the Hilbert series of a graded commutative algebra finitely generated over a field are three strongly related notions which measure the growth of the dimension of the homog ...
of a polynomial ring in ''k'' + 1 variables is the numerical polynomial
.
References
Algebra
*
*
Algebraic topology
*
*
*
*
Further reading
*
{{DEFAULTSORT:Integer-Valued Polynomial
Polynomials
Number theory
Commutative algebra
Ring theory