Inmathematics, aninteger-valued polynomial (also known as anumerical polynomial)${\displaystyle P(t)}$ is apolynomial whose value${\displaystyle P(n)}$ is aninteger for every integern. Every polynomial with integercoefficients is integer-valued, but the converse is not true. For example, the polynomial

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

takes on integer values whenevert is an integer. That is because one oft and${\displaystyle t+1}$ must be aneven number. (The values this polynomial takes are thetriangular numbers.)

Integer-valued polynomials are objects of study in their own right in algebra, and frequently appear inalgebraic topology.^[1]

The class of integer-valued polynomials was described fully byGeorge Pólya (1915). Inside thepolynomial ring${\displaystyle \mathbb{Q}[t]}$ of polynomials withrational number coefficients, thesubring of integer-valued polynomials is afree abelian group. It has asbasis the polynomials

${\displaystyle P_k(t)=t(t-1)\cdots (t-k+1)/k!}$

for${\displaystyle k=0,1,2,\dots }$, i.e., thebinomial coefficients. In other words, every integer-valued polynomial can be written as an integerlinear combination of binomial coefficients in exactly one way. The proof is by the method ofdiscrete 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).

Integer-valued polynomials may be used effectively to solve questions about fixed divisors of polynomials. For example, the polynomialsP with integer coefficients that always take on even number values are just those such that${\displaystyle P/2}$ 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 asSchinzel's hypothesis H and theBateman–Horn conjecture, it is a matter of basic importance to understand the case whenP has no fixed prime divisor (this has been calledBunyakovsky's property^{[citation needed]}, afterViktor Bunyakovsky). By writingP in terms of the binomial coefficients, we see the highest fixed prime divisor is also the highest primecommon factor of the coefficients in such a representation. So Bunyakovsky's property is equivalent to coprime coefficients.

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

${\displaystyle n(n^2+2)}$

is divisible by 3, which follows from the representation

${\displaystyle n(n^2+2)=6(\genfrac{}{}{0ex}{}{n}{3})+6(\genfrac{}{}{0ex}{}{n}{2})+3(\genfrac{}{}{0ex}{}{n}{1})}$

with respect to the binomial basis, where the highest common factor of the coefficients—hence the highest fixed divisor of${\displaystyle n(n^2+2)}$—is 3.

Numerical polynomials can be defined over other rings and fields, in which case the integer-valued polynomials above are referred to asclassical numerical polynomials.^{[citation needed]}

TheK-theory ofBU(n) is numerical (symmetric) polynomials.

TheHilbert polynomial of a polynomial ring ink + 1 variables is the numerical polynomial${\displaystyle (\genfrac{}{}{0ex}{}{t+k}{k})}$.

• Cahen, Paul-Jean; Chabert, Jean-Luc (1997),Integer-valued polynomials, Mathematical Surveys and Monographs, vol. 48, Providence, RI:American Mathematical Society,MR 1421321
• Pólya, George (1915), "Über ganzwertige ganze Funktionen",Palermo Rend. (in German),40:1–16,doi:10.1007/BF03014836,ISSN 0009-725X,JFM 45.0655.02

• Baker, Andrew; Clarke, Francis; Ray, Nigel; Schwartz, Lionel (1989), "On the Kummer congruences and the stable homotopy ofBU",Transactions of the American Mathematical Society,316 (2):385–432,doi:10.2307/2001355,JSTOR 2001355,MR 0942424

• Ekedahl, Torsten (2002),"On minimal models in integral homotopy theory",Homology, Homotopy and Applications,4 (2):191–218,arXiv:math/0107004,doi:10.4310/hha.2002.v4.n2.a9,MR 1918189,Zbl 1065.55003

• Elliott, Jesse (2006). "Binomial rings, integer-valued polynomials, and λ-rings".Journal of Pure and Applied Algebra.207 (1):165–185.doi:10.1016/j.jpaa.2005.09.003.MR 2244389.

• Hubbuck, John R. (1997), "Numerical forms",Journal of the London Mathematical Society, Series 2,55 (1):65–75,doi:10.1112/S0024610796004395,MR 1423286

• Narkiewicz, Władysław (1995).Polynomial mappings. Lecture Notes in Mathematics. Vol. 1600. Berlin:Springer-Verlag.ISBN 3-540-59435-3.ISSN 0075-8434.Zbl 0829.11002.

• Polynomials
• Number theory
• Commutative algebra
• Ring theory

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