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 Template:Harvs. Inside the polynomial ring $\mathbb{Q}[t]$ of polynomials with rational number coefficients, the subring of integer-valued polynomials is a free abelian group. It has as basis the polynomials

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

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 propertyTemplate:Citation needed, 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

$n(n^2+2)=6(\genfrac{}{}{0ex}{}{n}{3})+6(\genfrac{}{}{0ex}{}{n}{2})+3(\genfrac{}{}{0ex}{}{n}{1})$

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.Template:Citation needed

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 $(\genfrac{}{}{0ex}{}{t+k}{k})$.

References

Algebra

• Template:Citation
• Template:Citation

Algebraic topology

• Template:Citation

• Template:Citation

• Template:Cite journal

• Template:Citation

Further reading

• Template:Cite book