# Multiplicative sequence

In mathematics, a **multiplicative sequence** or ** m-sequence** is a sequence of polynomials associated with a formal group structure. They have application in the cobordism ring in algebraic topology.

## DefinitionEdit

Let *K*_{n} be polynomials over a ring *A* in indeterminates *p*_{1},... weighted so that *p*_{i} has weight *i* (with *p*_{0} = 1) and all the terms in *K*_{n} have weight *n* (so that *K*_{n} is a polynomial in *p*_{1}, ..., *p*_{n}). The sequence *K*_{n} is *multiplicative* if an identity

implies

In other words, is required to be an endomomorphism of the multiplicative monoid .

The power series

is the *characteristic power series* of the *K*_{n}. A multiplicative sequence is determined by its characteristic power series *Q*(*z*), and every power series with constant term 1 gives rise to a multiplicative sequence.

To recover a multiplicative sequence from a characteristic power series *Q*(*z*) we consider the coefficient of *z*^{j} in the product

for any *m* > *j*. This is symmetric in the *β*_{i} and homogeneous of weight *j*: so can be expressed as a polynomial *K*_{j}(*p*_{1}, ..., *p*_{j}) in the elementary symmetric functions *p* of the *β*. Then *K*_{j} defines a multiplicative sequence.

## ExamplesEdit

As an example, the sequence *K*_{n} = *p*_{n} is multiplicative and has characteristic power series 1 + *z*.

Consider the power series

where *B*_{k} is the *k*-th Bernoulli number. The multiplicative sequence with *Q* as characteristic power series is denoted *L*_{j}(*p*_{1}, ..., *p*_{j}).

The multiplicative sequence with characteristic power series

is denoted *A*_{j}(*p*_{1},...,*p*_{j}).

The multiplicative sequence with characteristic power series

is denoted *T*_{j}(*p*_{1},...,*p*_{j}): these are the *Todd polynomials*.

## GenusEdit

The *genus* of a multiplicative sequence is a ring homomorphism, from the cobordism ring of smooth oriented compact manifolds to another ring, usually the ring of rational numbers.

For example, the Todd genus is associated to the Todd polynomials with characteristic power series .

## ReferencesEdit

- Hirzebruch, Friedrich (1995) [1978].
*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.). Berlin: Springer-Verlag. ISBN 3-540-58663-6. Zbl 0843.14009.