Editing Integer-valued polynomial

In [[mathematics]], an '''integer-valued polynomial''' (also known as a '''numerical polynomial''') <math>P(t)</math> is a [[polynomial]] whose value <math>P(n)</math> is an [[integer]] for every integer ''n''. Every polynomial with integer [[coefficient]]s is integer-valued, but the converse is not true. For example, the polynomial

:<math> P(t) = \frac{1}{2} t^2 + \frac{1}{2} t=\frac{1}{2}t(t+1)</math>

takes on integer values whenever ''t'' is an integer.  That is because one of ''t'' and <math>t + 1</math> must be an [[even number]]. (The values this polynomial takes are the [[triangular number]]s.)

Integer-valued polynomials are objects of study in their own right in algebra, and frequently appear in [[algebraic topology]].<ref>{{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.&nbsp;213–214.</ref>

==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]] <math>\Q[t]</math> of polynomials with [[rational number]] coefficients, the [[subring]] of integer-valued polynomials is a [[free abelian group]]. It has as [[basis (linear algebra)|basis]] the polynomials

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

for <math>k = 0,1,2, \dots</math>, i.e., the [[binomial coefficient]]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 [[Difference operator|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 <math>P/2</math> 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 <math>n</math> and <math>n^2 + 2</math> violates this condition at <math>p = 3</math>: for every <math>n</math> the product

:<math>n(n^2 + 2)</math>

is divisible by 3, which follows from the representation

:<math> n(n^2 + 2) = 6 \binom{n}{3} + 6 \binom{n}{2} + 3 \binom{n}{1} </math>

with respect to the binomial basis, where the highest common factor of the coefficients&mdash;hence the highest fixed divisor of <math>n(n^2+2)</math>&mdash;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 [[topological K-theory|K-theory]] of [[Classifying space for U(n)|BU(''n'')]] is numerical (symmetric) polynomials.

The [[Hilbert polynomial]] of a polynomial ring in ''k''&nbsp;+&nbsp;1 variables is the numerical polynomial <math>\binom{t+k}{k}</math>.

==References==
{{Reflist}}

===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=[[London Mathematical Society#Publications|Journal of the London Mathematical Society]]  |series=Series 2
|volume=55
|year=1997
|issue=1
|pages=65–75
|doi=10.1112/S0024610796004395
|mr=1423286 
}}

==Further reading==
* {{cite book | last=Narkiewicz | first=Władysław | title=Polynomial mappings | chapter=Fully invariant sets for polynomial mappings | series=Lecture Notes in Mathematics | volume=1600 | location=Berlin | publisher=[[Springer-Verlag]] | year=1995 | pages=67–109 | doi=10.1007/BFb0076896 | isbn=3-540-59435-3 | issn=0075-8434 | zbl=0829.11002 }}

{{DEFAULTSORT:Integer-Valued Polynomial}}
[[Category:Polynomials]]
[[Category:Number theory]]
[[Category:Commutative algebra]]
[[Category:Ring theory]]