Integer-valued polynomial

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title: "Integer-valued polynomial" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["polynomials", "number-theory", "commutative-algebra", "ring-theory"] topic_path: "science/mathematics" source: "https://en.wikipedia.org/wiki/Integer-valued_polynomial" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

In mathematics, an integer-valued polynomial (also known as a numerical polynomial) P(t) is a polynomial whose value P(n) is an integer for every integer n. Every polynomial with integer coefficients is integer-valued, but the converse is not true. For example, the polynomial

: P(t) = \frac{1}{2} t^2 + \frac{1}{2} t=\frac{1}{2}t(t+1)

takes on integer values whenever t is an integer. That is because one of t and t + 1 must be an even number. (The values this polynomial takes are the triangular numbers.)

Integer-valued polynomials are objects of study in their own right in algebra, and frequently appear in algebraic topology.

Classification

The class of integer-valued polynomials was described fully by . Inside the polynomial ring \Q[t] of polynomials with rational number coefficients, the subring of integer-valued polynomials is a free abelian group. It has as basis the polynomials

:P_k(t) = t(t-1)\cdots (t-k+1)/k!

for k = 0,1,2, \dots, i.e., the binomial coefficients. 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 P/2 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, 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). By writing P in terms of the binomial coefficients, we see the highest fixed prime divisor is also the highest prime common factor of the coefficients in such a representation. So Bunyakovsky's property is equivalent to coprime coefficients.

As an example, the pair of polynomials n and n^2 + 2 violates this condition at p = 3: for every n the product

:n(n^2 + 2)

is divisible by 3, which follows from the representation

: n(n^2 + 2) = 6 \binom{n}{3} + 6 \binom{n}{2} + 3 \binom{n}{1}

with respect to the binomial basis, where the highest common factor of the coefficients—hence the highest fixed divisor of n(n^2+2)—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 of BU(n) is numerical (symmetric) polynomials.

The Hilbert polynomial of a polynomial ring in k + 1 variables is the numerical polynomial \binom{t+k}{k}.

References

Algebra

  • {{citation |last1=Cahen |first1=Paul-Jean |last2=Chabert |first2=Jean-Luc |title=Integer-valued polynomials |series=Mathematical Surveys and Monographs |volume=48 |publisher=American Mathematical Society |location=Providence, RI |year=1997 |mr=1421321

Algebraic topology

  • {{citation |first1=Andrew |last1= Baker |first2=Francis |last2= Clarke |first3=Nigel |last3= Ray |first4=Lionel |last4= Schwartz |title=On the Kummer congruences and the stable homotopy of BU |journal=Transactions of the American Mathematical Society |volume=316 |issue=2 |year=1989 |pages=385–432 |doi=10.2307/2001355 |jstor=2001355 |mr=0942424

  • {{citation |first=Torsten|last= Ekedahl |title=On minimal models in integral homotopy theory |journal=Homology, Homotopy and Applications |volume=4 |issue=2 |year=2002 |pages=191–218 |url=http://projecteuclid.org/euclid.hha/1139852462 |doi=10.4310/hha.2002.v4.n2.a9 |zbl = 1065.55003 |mr=1918189|doi-access=free |arxiv=math/0107004

  • {{citation |first=John R.|last= Hubbuck |title=Numerical forms |journal=Journal of the London Mathematical Society |series=Series 2 |volume=55 |year=1997 |issue=1 |pages=65–75 |doi=10.1112/S0024610796004395 |mr=1423286

References

  1. Johnson, Keith. (2014). "Commutative Algebra: Recent Advances in Commutative Rings, Integer-Valued Polynomials, and Polynomial Functions". Springer.

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polynomialsnumber-theorycommutative-algebraring-theory