Todd Polynomials

2020 Mathematics Subject Classification: Primary: 57R [MSN][ZBL]

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

Let $$H(z;{\xi }_1,\dots ,{\xi }_s)=\prod _{i=1}^s\frac{z{\xi }_i}{1-\exp (-z{\xi }_i)}.$$ 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[edit]

• 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.