Related papers: Multivalued groups and Newton polyhedron

We define the monoidal category $(Poly_E,y,\triangleleft)$ of polynomials under composition in any category $E$ with finite limits, including both cartesian and vertical morphisms of polynomials, and generalize to this setting the Dirichlet…

In this article, we carry out the investigation for regular sequences of symmetric polynomials in the polynomial ring in three and four variable. Any two power sum element in $\mathbb{C}[x_1,x_2,...,x_n]$ for $n \geq 3$ always form a…

Let D be a domain with quotient field K and A a D-algebra. We call a polynomial with coefficients in K that maps every element of A to an element of A "integer-valued on A". For commutative A we also consider integer-valued polynomials in…

Let $p$ be a prime number. Every $n$-variable polynomial $f(\underline x)$ over a finite field of characteristic $p$ defines an Artin--Schreier--Witt tower of varieties whose Galois group is isomorphic to $\mathbb{Z}_p$. Our goal of this…

In this work, it is shown that if $A$ is an $n$-by-$n$ convexoid matrix (i.e., its field of values coincides with the convex hull of its eigenvalues), then the field of any $(n-1)$-by-$(n-1)$ principal submatrix of $A$ is inscribed in the…

In combinatorics, P\'{o}lya's Enumeration Theorem is a powerful tool for solving a wide range of counting problems, including the enumeration of groups, graphs, and chemical compounds. In this paper, we present an extension of P\'{o}lya's…

A polyuble of a Manin triple can be regarded as the ``$n$-th power'' of it, which plays an important rule in the study of Poisson geometry, mathematical physics and Lie theory. In this paper, we first construct an isomorphism between the…

We analyze the structure of the algebra N of symmetric polynomials in non-commuting variables in so far as it relates to its commutative counterpart. Using the "place-action" of the symmetric group, we are able to realize the latter as the…

We give an elementary introduction to some recent polyhedral techniques for understanding and solving systems of multivariate polynomial equations. We provide numerous concrete examples and illustrations, and assume no background in…

A sequence $\sigma$ of $p$ non-negative integers is unigraphic if it is the degree sequence of exactly one graph, up to isomorphism. A polyhedral graph is a $3$-connected, planar graph. We investigate which sequences are unigraphic with…

We give a simple matrix-based proof of congruence equations modulo a prime $p$ involving sums of binomial coefficients appearing in Pascal's triangle. These equations can be used to construct some groups of exponent $p^n$. These groups, as…

A sharp bound is derived for the nilpotency class of a regular p-group in terms of its coexponent, and is used to show that the number of groups of order p^n with a given fixed coexponent, is independent of n, for p and n sufficiently…

In this paper we calculate the Ehrhart's polynomial associated with a 2-dimensional regular polytope (i.e. equilateral triangles) in $\mathbb Z^3$. The polynomial takes a relatively simple form in terms of the coordinates of the vertices of…

For each positive integer n this paper considers a one-parameter family of complex-valued measures on the symmetric group S_n, depending on a complex parameter z. For parameter values z=p^f a prime power, this measure describes splitting…

We introduce the Z-polynomial of a matroid, which we define in terms of the Kazhdan-Lusztig polynomial. We then exploit a symmetry of the Z-polynomial to derive a new recursion for Kazhdan-Lusztig coefficients. We solve this recursion,…

We consider cubic polynomials f(z)=z^3+az+b defined over the function field C(L), with a marked point of period N and multiplier L. In the case N=1, there are infinitely many such objects, and in the case N>2, only finitely many. The case…

Let $L_n$ be the free metabelian Leibniz algebra generated by the set $X_n=\{x_1,\ldots,x_n\}$ over a field $K$ of characteristic zero. This is the free algebra of rank $n$ in the variety of solvable of class $2$ Leibniz algebras. We call…

Let R be a commutative ring and let n,m be two positive integers. The symmetric group on n letters acts diagonally on the ring of polynomials in nxm variables with coefficients in R. The subrings of invariants for this action is called the…

Newton's identities provide a way to express elementary symmetric polynomials in terms of power polynomials over fields of characteristic zero. In this article, we study the failure of this relation in positive characteristic and what can…

A classic theorem by Steinitz states that a graph G is realizable by a convex polyhedron if and only if G is 3-connected planar. Zonohedra are an important subclass of convex polyhedra having the property that the faces of a zonohedron are…