Integer-valued polynomial
In mathematics, an integer-valued polynomial (also known as a numerical polynomial) P(t) is a polynomial whose value 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
- 12t2+12t=12t(t+1){displaystyle {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 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]
Contents
1 Classification
2 Fixed prime divisors
3 Other rings
4 Applications
5 References
5.1 Algebra
5.2 Algebraic topology
6 Further reading
Classification
The class of integer-valued polynomials was described fully by Pólya (1915). 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
Pk(t) = t(t − 1)...(t − k + 1)/k!
for k = 0,1,2, ..., 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 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 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.
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.[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 (t+kk){displaystyle {binom {t+k}{k}}}.
References
^ Johnson, Keith (2014), "Stable homotopy theory, formal group laws, and integer-valued polynomials", in Fontana, Marco; Frisch, Sophie; Glaz, Sarah, Commutative Algebra: Recent Advances in Commutative Rings, Integer-Valued Polynomials, and Polynomial Functions, Springer, pp. 213–224, ISBN 9781493909254.mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:"""""""'""'"}.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}. See in particular pp. 213–214.
Algebra
Cahen, P-J.; Chabert, J-L. (1997), Integer-valued polynomials, Mathematical Surveys and Monographs, 48, Providence, RI: American Mathematical Society
Pólya, G. (1915), "Über ganzwertige ganze Funktionen", Palermo Rend. (in German), 40: 1–16, ISSN 0009-725X, JFM 45.0655.02
Algebraic topology
A. Baker; F. Clarke; N. Ray; L. Schwartz (1989), "On the Kummer congruences and the stable homotopy of BU", Trans. Amer. Math. Soc., Transactions of the American Mathematical Society, Vol. 316, No. 2, 316 (2): 385–432, doi:10.2307/2001355, JSTOR 2001355
T. Ekedahl (2002), "On minimal models in integral homotopy theory", Homology Homotopy Appl., 4 (2): 191–218, Zbl 1065.55003
J. Hubbuck (1997), "Numerical forms", J. London Math. Soc., Series 2, 55 (1): 65–75, doi:10.1112/S0024610796004395
Further reading
Narkiewicz, Władysław (1995). Polynomial mappings. Lecture Notes in Mathematics. 1600. Berlin: Springer-Verlag. ISBN 3-540-59435-3. ISSN 0075-8434. Zbl 0829.11002.