- Integer-valued polynomial
- Integer-valued function
- Polynomial
- List of polynomial topics
- Algebraic integer
- Schinzel's hypothesis H
- Binomial coefficient
- Elementary symmetric polynomial
- Cyclotomic polynomial
- Bunyakovsky conjecture
Integer-valued polynomial GudangMovies21 Rebahinxxi LK21
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
= Algebra
=Cahen, Paul-Jean; Chabert, Jean-Luc (1997), Integer-valued polynomials, Mathematical Surveys and Monographs, vol. 48, Providence, RI: American Mathematical Society, MR 1421321
Pólya, George (1915), "Über ganzwertige ganze Funktionen", Palermo Rend. (in German), 40: 1–16, ISSN 0009-725X, JFM 45.0655.02
= Algebraic topology
=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
Further reading
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.
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