Numerical polynomial

From Wikipedia, the free encyclopedia

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.

1 Classification
2 Other rings
3 Applications
4 References
- 4.1 Algebra
- 4.2 Algebraic topology

[edit] 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).

[edit] 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.

[edit] 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 $\binom{t+k}{k}$ .

[edit] References

[edit] Algebra

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