Multiplicative sequence: Difference between revisions

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==Definition==

Let ''K''_''n'' be polynomials over a [[ring (mathematics)|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'' (soin thatparticular ''K''_''n'' is a polynomial in ''p''₁, ..., ''p''_''n''). The sequence ''K''_''n'' is ''multiplicative'' if anthe identitymap

:<math>\sum_i p_iK: z^i\sum_{n = 0}^\sum_iinfty p'_i zq_nz^i n\cdotmapsto \sum_isum_{n=0}^\infty p''_iK_n(q_1,\cdots, q_n)z^in </math>

In other words, <math>p\mapsto K(p)</math> is required to be an [[endomorphism]] of the multiplicative [[monoid]] <math>(A[x_1, x_2,\cdots][X[z]],\cdot)</math>, where <math>q_n\in A[x_1, x_2,\cdots]</math>.

implies

:<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 [[endomorphism]] of the multiplicative [[monoid]] <math>(A[[X]],\cdot)</math>.

The [[power series]]

:<math>K(1+z) = \sum K_n(1,0,\ldots,0) z^n </math>

is the ''characteristic power series'' of the&nbsp;''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.

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==References==

* {{cite book | zbl=0843.14009 | last=Hirzebruch | first=Friedrich | authorlink=Friedrich Hirzebruch | title=Topological methods in algebraic geometry | others=Translation from the German and appendix one by R. L. E. Schwarzenberger. Appendix two by A. Borel | edition=Reprint of the 2nd, corr. print. of the 3rd | origyearorig-date=1978 | series=Classics in Mathematics | ___location=Berlin | publisher=[[Springer-Verlag]] | year=1995 | isbn=3-540-58663-6 }}

[[Category:Polynomials]]