integer-valued polynomial

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.{{citation|title=Commutative Algebra: Recent Advances in Commutative Rings, Integer-Valued Polynomials, and Polynomial Functions|editor1-first=Marco|editor1-last=Fontana|editor2-first=Sophie|editor2-last=Frisch|editor3-first=Sarah|editor3-last=Glaz|editor3-link=Sarah Glaz|publisher=Springer|year=2014|isbn=9781493909254|contribution=Stable homotopy theory, formal group laws, and integer-valued polynomials|first=Keith|last=Johnson|url=https://books.google.com/books?id=ZZEpBAAAQBAJ&pg=PA213|pages=213–224}}. See in particular pp. 213–214.

Classification

The class of integer-valued polynomials was described fully by {{harvs|txt|first=George|last=Pólya|authorlink=George Pólya|year=1915}}. 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{{Citation needed|date=January 2013}}, 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.{{citation needed|date=April 2012}}

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

}}

{{citation | last=Pólya | first=George | authorlink=George Pólya | title=Über ganzwertige ganze Funktionen | language=German | journal=Palermo Rend. | volume=40 | pages=1–16 | year=1915 | doi=10.1007/BF03014836 | issn=0009-725X | jfm=45.0655.02 }}

=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

}}

{{cite journal | last=Elliott | first=Jesse | title=Binomial rings, integer-valued polynomials, and λ-rings | journal=Journal of Pure and Applied Algebra | volume=207 | issue=1 | year=2006 | doi=10.1016/j.jpaa.2005.09.003 | pages=165–185|mr=2244389| doi-access= }}

{{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

}}