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

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

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.

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, for 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 n² + 2 violates this condition at p = 3: for every n the product

n(n² + 2)

is divisible by 3. Consequently there cannot be infinitely many prime pairs n and n² + 2. The divisibility is attributable to the alternate representation

n(n + 1)(n − 1) + 3n.

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 $\binom{t+k}{k}$.

Further reading

• . See in particular pp. 213–214.