Abelian Functions for Cyclic Trigonal Curves of Genus Four

We discuss the theory of generalized Weierstrass $\sigma$ 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 $y^3=x^5+\lambda_4 x^4 +\lambda_3 x^3+\lambda_2 x^2 +\lambda_1 x+\lambda_0$ 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

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

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

Polytopes, Hopf algebras and Quasi-symmetric functions

In this paper we use the technique of Hopf algebras and quasi-symmetric functions to study the combinatorial polytopes. Consider the free abelian group $\mathcal{P}$ 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. $(\mathcal{P},\times)$ is a commutative associative bigraded ring of polynomials, and $\mathcal{RP}=(\mathbb Z\varnothing\oplus\mathcal{P},\divideontimes)$ is a commutative associative threegraded ring of polynomials. The ring $\mathcal{RP}$ has the structure of a graded Hopf algebra. It turns out that $\mathcal{P}$ has a natural Hopf comodule structure over $\mathcal{RP}$. Faces operators $d_k$ that send a polytope to the sum of all its $(n-k)$-dimensional faces define on both rings the Hopf module structures over the universal Leibnitz-Hopf algebra $\mathcal{Z}$. This structure gives a ring homomorphism \R\to\Qs\otimes\R, where $\R$ is $\mathcal{P}$ or $\mathcal{RP}$. Composing this homomorphism with the characters $P^n\to\alpha^n$ of $\mathcal{P}$, $P^n\to\alpha^{n+1}$ of $\mathcal{RP}$, 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 $F$ 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 $\mathbb Q$, and find the relations between the images, the homomorphisms and the Hopf comodule structures. For each homomorphism $f,\;f_{\mathcal{RP}}$, and \F the images of two polytopes coincide if and only if they have equal flag $f$-vectors. Therefore algebraic structures on the images give the information about flag $f$-vectors of polytopes.Comment: 61 page