370 research outputs found

    Abelian Functions for Cyclic Trigonal Curves of Genus Four

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    We discuss the theory of generalized Weierstrass σ\sigma and \wp functions defined on a trigonal curve of genus four, following earlier work on the genus three case. The specific example of the "purely trigonal" (or "cyclic trigonal") curve y3=x5+λ4x4+λ3x3+λ2x2+λ1x+λ0y^3=x^5+\lambda_4 x^4 +\lambda_3 x^3+\lambda_2 x^2 +\lambda_1 x+\lambda_0 is discussed in detail, including a list of some of the associated partial differential equations satisfied by the \wp functions, and the derivation of an addition formulae.Comment: 23 page

    Hyperelliptic addition law

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    We construct an explicit form of the addition law for hyperelliptic Abelian vector functions \wp and \wp'. The functions \wp and \wp' form a basis in the field of hyperelliptic Abelian functions, i.e., any function from the field can be expressed as a rational function of \wp and \wp'.Comment: 18 pages, amslate

    K^*(BG) rings for groups G=G38,...,G41G=G_{38},...,G_{41} of order 32

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    B. Schuster \cite{SCH1} proved that the modmod 2 Morava KK-theory K(s)(BG)K(s)^*(BG) is evenly generated for all groups GG of order 32. For the four groups GG with the numbers 38, 39, 40 and 41 in the Hall-Senior list \cite{H}, the ring K(2)(BG)K(2)^*(BG) has been shown to be generated as a K(2)K(2)^*-module by transferred Euler classes. In this paper, we show this for arbitrary ss and compute the ring structure of K(s)(BG)K(s)^*(BG). Namely, we show that K(s)(BG)K(s)^*(BG) is the quotient of a polynomial ring in 6 variables over K(s)(pt)K(s)^*(pt) by an ideal for which we list explicit generators.Comment: 23 page

    Polytopes, Hopf algebras and Quasi-symmetric functions

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    In this paper we use the technique of Hopf algebras and quasi-symmetric functions to study the combinatorial polytopes. Consider the free abelian group P\mathcal{P} generated by all combinatorial polytopes. There are two natural bilinear operations on this group defined by a direct product ×\times and a join \divideontimes of polytopes. (P,×)(\mathcal{P},\times) is a commutative associative bigraded ring of polynomials, and RP=(ZP,)\mathcal{RP}=(\mathbb Z\varnothing\oplus\mathcal{P},\divideontimes) is a commutative associative threegraded ring of polynomials. The ring RP\mathcal{RP} has the structure of a graded Hopf algebra. It turns out that P\mathcal{P} has a natural Hopf comodule structure over RP\mathcal{RP}. Faces operators dkd_k that send a polytope to the sum of all its (nk)(n-k)-dimensional faces define on both rings the Hopf module structures over the universal Leibnitz-Hopf algebra Z\mathcal{Z}. This structure gives a ring homomorphism \R\to\Qs\otimes\R, where R\R is P\mathcal{P} or RP\mathcal{RP}. Composing this homomorphism with the characters PnαnP^n\to\alpha^n of P\mathcal{P}, Pnαn+1P^n\to\alpha^{n+1} of RP\mathcal{RP}, and with the counit we obtain the ring homomorphisms f\colon\mathcal{P}\to\Qs[\alpha], f_{\mathcal{RP}}\colon\mathcal{RP}\to\Qs[\alpha], and \F^*:\mathcal{RP}\to\Qs, where FF is the Ehrenborg transformation. We describe the images of these homomorphisms in terms of functional equations, prove that these images are rings of polynomials over Q\mathbb Q, and find the relations between the images, the homomorphisms and the Hopf comodule structures. For each homomorphism f,  fRPf,\;f_{\mathcal{RP}}, and \F the images of two polytopes coincide if and only if they have equal flag ff-vectors. Therefore algebraic structures on the images give the information about flag ff-vectors of polytopes.Comment: 61 page