Integer-valued polynomial

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.^[1]

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).

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^{[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 {\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.

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.^{[citation needed]}

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}}} .

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• Pólya, George (1915), "Über ganzwertige ganze Funktionen", Palermo Rend. (in German), 40: 1–16, doi:10.1007/BF03014836, ISSN 0009-725X, JFM 45.0655.02

• Baker, Andrew; Clarke, Francis; Ray, Nigel; Schwartz, Lionel (1989), "On the Kummer congruences and the stable homotopy of BU", Transactions of the American Mathematical Society, 316 (2): 385–432, doi:10.2307/2001355, JSTOR 2001355, MR 0942424

• Ekedahl, Torsten (2002), "On minimal models in integral homotopy theory", Homology, Homotopy and Applications, 4 (2): 191–218, arXiv:math/0107004, doi:10.4310/hha.2002.v4.n2.a9, MR 1918189, Zbl 1065.55003

• Elliott, Jesse (2006). "Binomial rings, integer-valued polynomials, and λ-rings". Journal of Pure and Applied Algebra. 207 (1): 165–185. doi:10.1016/j.jpaa.2005.09.003. MR 2244389.

• Hubbuck, John R. (1997), "Numerical forms", Journal of the London Mathematical Society, Series 2, 55 (1): 65–75, doi:10.1112/S0024610796004395, MR 1423286

• 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.