371 research outputs found

### 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 $G=G_{38},...,G_{41}$ 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

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