# Todd polynomials

2010 Mathematics Subject Classification: *Primary:* 57R [MSN][ZBL]

A sequence of polynomials with rational number coefficients associated with Todd classes.

Let $$ H(z; \xi_1,\ldots,\xi_s) = \prod_{i=1}^s \frac{z \xi_i}{1 - \exp(-z\xi_i)} \ . $$ The $m$-th Todd polynomial $T_m(c_1,\ldots,c_m)$ is defined by $T_m(\sigma_1,\ldots,\sigma_m)$ being the coefficient of $z^m$ in the power series expansion of $H(z; \xi_1,\ldots,\xi_m)$ where the $\sigma_i$ are the elementary symmetric functions of the $\xi_i$.

We have $T_1(c_1) = \frac12 c_1$, $T_2(c_1,c_2) = \frac1{12}(c_1^2 + c_2)$, $T_3(c_1,c_2,c_3) = \frac{1}{24}c_1c_2$.

The Todd polynomials are derived from the multiplicative sequence corresponding to the power series $t/(1-e^{-t})$.

## References

- Hirzebruch, Friedrich.
*Topological methods in algebraic geometry*, Classics in Mathematics. Translation from the German and appendix one by R. L. E. Schwarzenberger. Appendix two by A. Borel. Reprint of the 2nd, corr. print. of the 3rd ed. [1978] Berlin: Springer-Verlag (1995). ISBN 3-540-58663-6. Zbl 0843.14009.

**How to Cite This Entry:**

Todd polynomials.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Todd_polynomials&oldid=35553