Related papers: Representation of $m$-gonal forms over $\mathbb N_…
It is known that any $m$-gonal form of rank $n \ge 5$ is almost regular. On the other words, any $m$-gonal form of rank $n \ge 5$ represents every sufficiently large integer which is locally represented. In this article, we study the…
It is known that any $m$-gonal form of $\rank n \ge 5$ is almost regular. In this article, we study the sufficiently large integers which are represented by (almost regular) $m$-gonal forms of $\rank n \ge 6$.
In this paper, we study the set of positive integers that characterize the universality of $m$-gonal form.
In this paper, we show that for a fixed rank $n$, there are only finitely many $m$ for which there is a regular $m$-gonal form of rank $n$ and determine every type of the (generalized) regular $m$-gonal form for every sufficiently large…
In this article, we consider the rank of universal $m$-gonal forms for all sufficiently large $m$. Especially, we determine the minimal rank of universal $m$-gonal form and the maximal rank of kinds of proper universal $m$-gonal form.
In this paper, we classify the $(a_1,a_2,a_3,a_4,a_5)$ for which the universality of an $m$-gonal form $F_m(\mathbf x)$ having its first five coefficients as $(a_1,a_2,a_3,a_4,a_5)$ is characterized as the representability of positive…
In this paper, we consider mixed sums of generalized polygonal numbers. Specifically, we obtain a finiteness condition for universality of such sums; this means that it suffices to check representability of a finite subset of the positive…
In this paper, we show that for any $m$-gonal form $F_m(\mathbf x)$ with $m \ge 12$ which represents every positive integer up to $m-4$, by putting together only unary $m$-gonal form, we may complete an universal form.
In this paper we study universal quadratic polynomials which arise as sums of polygonal numbers. Specifically, we determine an asymptotic upper bound (as a function of $m$) on the size of the set $S_m\subset\mathbb{N}$ such that if a sum of…
In this paper, we consider representations of integers as sums of at most four distinct $m$-gonal numbers (allowing a fixed number of repeats of each polygonal number occurring in the sum). We show that the number of such representations…
In this paper, we consider universal sums of generalized polygonal numbers. Fixing $m\in\mathbb{N}_{\geq 3}$, we show two finiteness theorems for universal sums of generalized polygonal numbers whose inputs have a restricted number $L$ of…
The orthogonal groups are a series of simple Lie groups associated to symmetric bilinear forms. There is no analogous series associated to symmetric trilinear forms. We introduce an infinite dimensional group-like object that can be viewed…
This is the second part of our study of epimorphisms with source a thick generalized $m$-gon and target a thin generalized $m$-gon. We classify the case $m = 8$ when the polygons are finite (in the first part [15] we handled the cases $m =…
In math.RT/0302174 we developed a framework to study representations of groups of the form $G((t))$, where $G$ is an algebraic group over a local field $K$. The main feature of this theory is that natural representations of groups of this…
The aim of this paper is to unveil an unexpected relationship between the normal form of a polynomial with respect to a polynomial ideal and the more geometric concept of orthogonality. We present a new way to calculate the normal form of a…
In this paper, we show that certain sums of generalized $m$-gonal numbers represent every positive integer if and only if they represent every positive integer up to an explicit bound $C_m$, verifying a conjecture of Sun for sufficiently…
In this paper, for n a positve integer, we compute the number of n degree representations for a dihedral group G of order 2m, m \geq 3 and the dimensions of the corresponding spaces of G invariant bilinear forms over a complex field C. We…
In 2014, we determine the precise form of a continuous orthogonal form on a commutative real C$^*$-algebra. We also describe the general form of a (not-necessarily continuous) orthogonality preserving linear map between commutative unital…
For a fixed integer $k \ge 0$, consider representations of positive integers as sums of binomial coefficients of the form $\binom{n}{k}$. While exact minimal bounds for the number of required summands are known only in a few low-dimensional…
Let M be an oriented complete hyperbolic n-manifold of finite volume. Using the definition of volume of a representation previously given by the authors in [BucherBurgerIozzi2013] we show that the volume of a representation of the…