Numerical polynomial

In mathematics, a numerical polynomial is a polynomial with rational coefficients that takes integer values on integers. They are also called integer-valued polynomials.

They are objects of study in their own right in algebra, and are frequently used in algebraic topology.

Classification

Polynomials with integer coefficients are numerical polynomials, but they are not the only ones. Binomial coefficients, thought of as rational polynomials, are also numerical polynomials. For instance,

$\binom{n}{2} = \frac{n(n-1)}{2} = \frac{1}{2}n^2 - \frac{1}{2}n$

is numerical but does not have integer coefficients.

In fact, binomial coefficients generate numerical polynomials: every numerical polynomial is a unique integer linear combination of binomial coefficients, via the discrete Taylor series: binomial coefficients are numerical 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).

See binomial coefficient#Binomial coefficients as a basis for the space of polynomials for more information.

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 $\binom{t+k}{k}$.

References

Algebra

• Cahen, P-J.; Chabert, J-L. (1997), Integer-valued polynomials, Mathematical Surveys and Monographs, 48, Providence, RI: American Mathematical Society 

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, http://projecteuclid.org/euclid.hha/1139852462 

• J. Hubbuck (1997), "Numerical forms", J. London Math. Soc. (2) 55 (1): 65–75, doi:10.1112/S0024610796004395