Английская Википедия:Integer-valued polynomial
In mathematics, an integer-valued polynomial (also known as a numerical polynomial) <math>P(t)</math> is a polynomial whose value <math>P(n)</math> 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
- <math> P(t) = \frac{1}{2} t^2 + \frac{1}{2} t=\frac{1}{2}t(t+1)</math>
takes on integer values whenever t is an integer. That is because one of t and <math>t + 1</math> 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
The class of integer-valued polynomials was described fully by Шаблон:Harvs. Inside the polynomial ring <math>\Q[t]</math> of polynomials with rational number coefficients, the subring of integer-valued polynomials is a free abelian group. It has as basis the polynomials
- <math>P_k(t) = t(t-1)\cdots (t-k+1)/k!</math>
for <math>k = 0,1,2, \dots</math>, 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 <math>P/2</math> 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 propertyШаблон: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 <math>n</math> and <math>n^2 + 2</math> violates this condition at <math>p = 3</math>: for every <math>n</math> the product
- <math>n(n^2 + 2)</math>
is divisible by 3, which follows from the representation
- <math> n(n^2 + 2) = 6 \binom{n}{3} + 6 \binom{n}{2} + 3 \binom{n}{1} </math>
with respect to the binomial basis, where the highest common factor of the coefficients—hence the highest fixed divisor of <math>n(n^2+2)</math>—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.Шаблон: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 <math>\binom{t+k}{k}</math>.
References
Algebra
Algebraic topology
Further reading
- ↑ Шаблон:Citation. See in particular pp. 213–214.
