In mathematics, an integer-valued polynomial (also known as a numerical polynomial) P ( t ) {\displaystyle P(t)} is a polynomial whose value P ( n ) {\displaystyle 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 t 2 + 1 2 t = 1 2 t ( t + 1 ) {\displaystyle 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 {\displaystyle 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 George Pólya (1915). Inside the polynomial ring Q [ t ] {\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
P k ( t ) = t ( t − 1 ) ⋯ ( t − k + 1 ) / k ! {\displaystyle P_{k}(t)=t(t-1)\cdots (t-k+1)/k!}
for k = 0 , 1 , 2 , … {\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).
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 {\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 n {\displaystyle n} and n 2 + 2 {\displaystyle n^{2}+2} violates this condition at p = 3 {\displaystyle p=3} : for every n {\displaystyle n} the product
n ( n 2 + 2 ) {\displaystyle n(n^{2}+2)}
is divisible by 3, which follows from the representation
n ( n 2 + 2 ) = 6 ( n 3 ) + 6 ( n 2 ) + 3 ( n 1 ) {\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 n ( n 2 + 2 ) {\displaystyle 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.
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 ( t + k k ) {\displaystyle {\binom {t+k}{k}}} .
References
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