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In [[mathematics]], a '''multiplicative sequence''' or '''''m''-sequence''' is a [[sequence]] of [[polynomial]]s associated with a formal [[group
==Definition==
Let ''K''<sub>''n''</sub> be polynomials over a [[ring (mathematics)|ring]] ''A'' in indeterminates ''p''<sub>1</sub>, ... weighted so that ''p''<sub>''i''</sub> has weight ''i'' (with ''p''<sub>0</sub> = 1) and all the terms in ''K''<sub>''n''</sub> have weight ''n'' (so that ''K''<sub>''n''</sub> is a polynomial in ''p''<sub>1</sub>, ..., ''p''<sub>''n''</sub>). The sequence ''K''<sub>''n''</sub> is ''multiplicative'' if an identity
:<math>\sum_i p_i z^i = \sum_i p'_i z^i \cdot \sum_i p''_i z^i </math>
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:<math>\sum_i K_i(p_1,\ldots,p_i) z^i = \sum_j K_j(p'_1,\ldots,p'_j) z^j \cdot \sum_k K_k(p''_1,\ldots,p''_k) z^k </math>
In other words, <math>p\mapsto K(p)</math> is required to be an
The [[power series]]
:<math>\sum K_n(1,0,\ldots,0) z^n </math>
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is the ''characteristic power series'' of the ''K''<sub>''n''</sub>. 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''<sup> ''j''</sup> in the product
:<math> \prod_{i=1}^m Q(\beta_i z) \ </math>
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==Examples==
As an example, the sequence ''K''<sub>''n''</sub> = ''p''<sub>''n''</sub> is multiplicative and has characteristic power series
Consider the power series
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{{main|Genus of a multiplicative sequence}}
The ''genus'' of a multiplicative sequence is a [[ring homomorphism]], from the [[cobordism|cobordism ring]] of smooth oriented [[compact manifold]]s to another
For example, the [[Todd genus]] is associated to the Todd polynomials with characteristic power series <math>\frac{z}{1-\exp(-z)}</math>.
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