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


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

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. 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


 * $$P_k(t) = t(t-1)\cdots (t-k+1)/k!$$

for $$k = 0,1,2, \dots$$, 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


 * $$ n(n^2 + 2) = 6 \binom{n}{3} + 6 \binom{n}{2} + 3 \binom{n}{1} $$

with respect to the binomial basis, where the highest common factor of the coefficients&mdash;hence the highest fixed divisor of $$n(n^2+2)$$&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.

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

Algebraic topology