Integer-valued polynomial

The class of integer-valued polynomials was described fully by George Pólya (1915). Inside the polynomial ring ${\displaystyle \mathbb {Q} [t]}$ of polynomials with rational number coefficients, the subring of integer-valued polynomials is a free abelian group. It has as basis the polynomials

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

for ${\displaystyle k=0,1,2,\dots }$, 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).

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 ${\displaystyle P/2}$ 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 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 ${\displaystyle n}$ 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

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

with respect to the binomial basis, where the highest common factor of the coefficients—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 as classical numerical polynomials.

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

• 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.