Integer-valued polynomial

In mathematics, an integer-valued polynomial (also known as a numerical polynomial) [math]\displaystyle{ P(t) }[/math] is a polynomial whose value [math]\displaystyle{ P(n) }[/math] 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

[math]\displaystyle{ 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]\displaystyle{ t + 1 }[/math] 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 George Pólya (1915). Inside the polynomial ring [math]\displaystyle{ \Q[t] }[/math] of polynomials with rational number coefficients, the subring of integer-valued polynomials is a free abelian group. It has as basis the polynomials

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

for [math]\displaystyle{ k = 0,1,2, \dots }[/math], 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 [math]\displaystyle{ 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, 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]\displaystyle{ n }[/math] and [math]\displaystyle{ n^2 + 2 }[/math] violates this condition at [math]\displaystyle{ p = 3 }[/math]: for every [math]\displaystyle{ n }[/math] the product

[math]\displaystyle{ n(n^2 + 2) }[/math]

is divisible by 3, which follows from the representation

[math]\displaystyle{ 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—hence the highest fixed divisor of [math]\displaystyle{ n(n^2+2) }[/math]—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 [math]\displaystyle{ \binom{t+k}{k} }[/math].

References

Algebra

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

Algebraic topology

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

• Ekedahl, Torsten (2002), "On minimal models in integral homotopy theory", Homology, Homotopy and Applications 4 (2): 191–218, doi:10.4310/hha.2002.v4.n2.a9, http://projecteuclid.org/euclid.hha/1139852462 

• 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. 

• Hubbuck, John R. (1997), "Numerical forms", Journal of the London Mathematical Society, 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.