Integer-valued polynomial explained

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)=

	1
	2

t²+

	1	t=
	2

	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.^[1]

Classification

\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, 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

and

n²+2

violates this condition at

p=3

: for every

the product

n(n²+2)

is divisible by 3, which follows from the representation

n(n²+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

n(n²⁺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.

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}

References

Algebraic topology

- - Elliott . Jesse . Binomial rings, integer-valued polynomials, and λ-rings . . 207 . 1 . 2006 . 10.1016/j.jpaa.2005.09.003 . 165–185. 2244389.

Notes and References

. See in particular pp. 213–214.