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    The characteristic polynomial of the Adams operators on graded connected Hopf algebras

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    The Adams operators Ψn\Psi_n on a Hopf algebra HH are the convolution powers of the identity of HH. We study the Adams operators when HH is graded connected. They are also called Hopf powers or Sweedler powers. The main result is a complete description of the characteristic polynomial (both eigenvalues and their multiplicities) for the action of the operator Ψn\Psi_n on each homogeneous component of HH. The eigenvalues are powers of nn. The multiplicities are independent of nn, and in fact only depend on the dimension sequence of HH. These results apply in particular to the antipode of HH (the case n=−1n=-1). We obtain closed forms for the generating function of the sequence of traces of the Adams operators. In the case of the antipode, the generating function bears a particularly simple relationship to the one for the dimension sequence. In case H is cofree, we give an alternative description for the characteristic polynomial and the trace of the antipode in terms of certain palindromic words. We discuss parallel results that hold for Hopf monoids in species and qq-Hopf algebras.Comment: 36 pages; two appendice

    Combinatorial Nullstellensatz modulo prime powers and the Parity Argument

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    We present new generalizations of Olson's theorem and of a consequence of Alon's Combinatorial Nullstellensatz. These enable us to extend some of their combinatorial applications with conditions modulo primes to conditions modulo prime powers. We analyze computational search problems corresponding to these kinds of combinatorial questions and we prove that the problem of finding degree-constrained subgraphs modulo 2d2^d such as 2d2^d-divisible subgraphs and the search problem corresponding to the Combinatorial Nullstellensatz over F2\mathbb{F}_2 belong to the complexity class Polynomial Parity Argument (PPA)
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