Integer-valued polynomial
Encyclopedia
In mathematics
, an integer-valued polynomial P(t) is a polynomial
taking an integer
value P(n) for every integer n. Certainly every polynomial with integer coefficient
s is integer-valued. There are simple examples to show that the converse is not true: for example the polynomial
giving the triangle numbers takes on integer values whenever t = n is an integer. That is because one out of n and n + 1 must be an even number.
In fact integer-valued polynomials can be described fully. Inside the polynomial ring
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
for k = 0,1,2, ... .
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 question when P has no fixed prime divisor (this has been called Bunyakovsky's property, for Viktor Bunyakovsky
). By writing P in terms of the basic polynomials, we see the highest fixed prime divisor is also the highest 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 n2 + 2 violates this condition at p = 3: for every n the product
is divisible by 3. Consequently there cannot be infinitely many prime pairs n and n2 + 2. The divisibility is attributable to the alternate representation
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, an integer-valued polynomial P(t) is a polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
taking an integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
value P(n) for every integer n. Certainly every polynomial with integer coefficient
Coefficient
In mathematics, a coefficient is a multiplicative factor in some term of an expression ; it is usually a number, but in any case does not involve any variables of the expression...
s is integer-valued. There are simple examples to show that the converse is not true: for example the polynomial
- t(t + 1)/2
giving the triangle numbers takes on integer values whenever t = n is an integer. That is because one out of n and n + 1 must be an even number.
In fact integer-valued polynomials can be described fully. Inside the polynomial ring
Polynomial ring
In mathematics, especially in the field of abstract algebra, a polynomial ring is a ring formed from the set of polynomials in one or more variables with coefficients in another ring. Polynomial rings have influenced much of mathematics, from the Hilbert basis theorem, to the construction of...
Q[t] of polynomials with rational number
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
coefficients, the subring
Subring
In mathematics, a subring of R is a subset of a ring, is itself a ring with the restrictions of the binary operations of addition and multiplication of R, and which contains the multiplicative identity of R...
of integer-valued polynomials is a free abelian group
Free abelian group
In abstract algebra, a free abelian group is an abelian group that has a "basis" in the sense that every element of the group can be written in one and only one way as a finite linear combination of elements of the basis, with integer coefficients. Hence, free abelian groups over a basis B are...
. It has as basis the polynomials
- Pk(t) = t(t − 1)...(t − k + 1)/k!
for k = 0,1,2, ... .
Fixed prime divisors
This concept 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 those expressed as sums of the basic polynomials, with even coefficients.In questions of prime number theory, such as Schinzel's hypothesis H
Schinzel's hypothesis H
In mathematics, Schinzel's hypothesis H is a very broad generalisation of conjectures such as the twin prime conjecture. It aims to define the possible scope of a conjecture of the nature that several sequences of the type...
and the Bateman–Horn conjecture, it is a matter of basic importance to understand the question when P has no fixed prime divisor (this has been called Bunyakovsky's property, for Viktor Bunyakovsky
Viktor Bunyakovsky
Viktor Yakovlevich Bunyakovsky was a Russian mathematician, member and later vice president of the Petersburg Academy of Sciences.He worked in theoretical mechanics and number theory , and is credited with an early discovery of the Cauchy-Schwarz inequality, proving it for the infinite dimensional...
). By writing P in terms of the basic polynomials, we see the highest fixed prime divisor is also the highest 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 n2 + 2 violates this condition at p = 3: for every n the product
- n(n2 + 2)
is divisible by 3. Consequently there cannot be infinitely many prime pairs n and n2 + 2. The divisibility is attributable to the alternate representation
- n(n + 1)(n − 1) + 3n.