# 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

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.[1]

## Classification

The class of integer-valued polynomials was described fully by Pólya (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)...(*t*−*k*+ 1)/*k*!

for *k* = 0,1,2, ..., 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

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 .

## References

- Johnson, Keith (2014), "Stable homotopy theory, formal group laws, and integer-valued polynomials", in Fontana, Marco; Frisch, Sophie; Glaz, Sarah (eds.),
*Commutative Algebra: Recent Advances in Commutative Rings, Integer-Valued Polynomials, and Polynomial Functions*, Springer, pp. 213–224, ISBN 9781493909254. See in particular pp. 213–214.

### Algebra

- Cahen, P-J.; Chabert, J-L. (1997),
*Integer-valued polynomials*, Mathematical Surveys and Monographs,**48**, Providence, RI: American Mathematical Society - Pólya, G. (1915), "Über ganzwertige ganze Funktionen",
*Palermo Rend.*(in German),**40**: 1–16, ISSN 0009-725X, JFM 45.0655.02

### Algebraic topology

- A. Baker; F. Clarke; N. Ray; L. Schwartz (1989), "On the Kummer congruences and the stable homotopy of
*BU*",*Trans. Amer. Math. Soc.*, Transactions of the American Mathematical Society, Vol. 316, No. 2,**316**(2): 385–432, doi:10.2307/2001355, JSTOR 2001355

- T. Ekedahl (2002), "On minimal models in integral homotopy theory",
*Homology Homotopy Appl.*,**4**(2): 191–218, Zbl 1065.55003

- J. Hubbuck (1997), "Numerical forms",
*J. London Math. Soc.*, Series 2,**55**(1): 65–75, doi:10.1112/S0024610796004395

## Further reading

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