Related papers: Pythagoras Numbers for Ternary Forms

We prove that the ring of integers in the totally real cubic subfield $K^{(49)}$ of the cyclotomic field $\mathbb{Q}(\zeta_7)$ has Pythagoras number equal to $4$. This is the smallest possible value for a totally real number field of odd…

Erd\H{o}s and Graham define $g(n) = n + \phi(n)$ and the iterated application $g_k(n) = g(g_{k-1}(n))$. They ask for solutions of $g_{k+r}(n) = 2 g_{k}(n)$ and observe $g_{k+2}(10) = 2 g_{k}(10)$ and $g_{k+2}(94) = 2 g_{k}(94)$. We show…

Ten values of 12-j symbols of the first kind published earlier are challenged by values calculated with an independent Python program. The program first implements a narrow class of square roots of rational numbers, utilizing Python's…

The Calabi-Yau spaces with SU(m) holonomy can be studied by the algebraic way through the integer lattice where one can construct the Newton reflexive polyhedra or the Berger graphs. Our conjecture is that the Berger graphs can be directly…

We develop the algebraic formalism of the formal ternary laws of C. Walter and we compare them to Buchstaber's 2-valued formal group laws. We also compute the "elementary" formal ternary laws (after inverting 2) using a computer program…

Classifications and representations are two main topics in the theory of quadratic forms. In this paper, we consider these topics of ternary quadratic forms. For a given squarefree integer $N$, first we give the classification of positive…

We study modular forms of some congruence subgroups. In this paper, we treat the cases level is 2-power, 3-power or 5. Structures of graded rings and many identities of infinite sum or infinite product are given. Theory of rational (1/3,…

We consider symmetric tensors of format $2 \times 2 \times 2$ and $2 \times 2 \times 2 \times 2$ over prime fields. Using computer algebra we compute the canonical forms of these tensors. For $2 \times 2 \times 2$ symmetric tensors, we…

Using modular forms we determine formulas for the number of representations of a positive integer by diagonal octonary quadratic forms with coefficients $1$, $2$, $3$ or $6$.

In this paper, I use Siegel-Weil formula and Kudla matching principle to prove some interesting identities between representation number (of ternary quadratic space) and the degree of Heegner divisors.

For each natural number $m\ge 3$, let $P_m(x)$ denote the generalized $m$-gonal number $\frac{(m-2)x^2-(m-4)x}{2}$ with $x\in\mathbb{Z}$. In this paper, with the help of the congruence theta function, we establish conditions on $a$, $b$,…

We investigate the $p$-adic valuation of Weil sums of the form $W_{F,d}(a)=\sum_{x \in F} \psi(x^d -a x)$, where $F$ is a finite field of characteristic $p$, $\psi$ is the canonical additive character of $F$, the exponent $d$ is relatively…

In this paper we solve the ternary Piatetski-Shapiro inequality with prime numbers of a special form. More precisely we show that, for any fixed $1<c<\frac{427}{400}$, every sufficiently large positive number $N$ and a small constant…

Let $S$ be a finite set of points in the plane that are in convex position. We present an algorithm that constructs a plane $\frac{3+4\pi}{3}$-spanner of $S$ whose vertex degree is at most 3. Let $\Lambda$ be the vertex set of a finite…

The purpose of this paper is to construct a crepant resolution of quotient singularities by trihedral groups ( finite subgroups of SL(3,C) of certain type ), and prove that each Euler number of the minimal model is equal to the number of…

We present a simple construction of an acute set of size $2^{d-1}+1$ in $\mathbb{R}^d$ for any dimension $d$. That is, we explicitly give $2^{d-1}+1$ points in the $d$-dimensional Euclidean space with the property that any three points form…

We study bounded width algebras which are minimal in the sense that every proper reduct does not have bounded width. We show that minimal bounded width algebras can be arranged into a pseudovariety with one basic ternary operation. We…

A famous theorem of Hilbert from 1888 states that a positive semidefinite (psd) real form is a sum of squares (sos) of real forms if and only if $n=2$ or $d=1$ or $(n,2d)=(3,4)$, where $n$ is the number of variables and $2d$ the degree of…

We enumerate complex curves on toric surfaces of any given degree and genus, having a single cusp and nodes as their singularities, and matching appropriately many point constraints. The solution is obtained via tropical enumerative…

We introduce positive Gorenstein ideals. These are Gorenstein ideals in the graded ring $\RR[x]$ with socle in degree 2d, which when viewed as a linear functional on $\RR[x]_{2d}$ is nonnegative on squares. Equivalently, positive Gorenstein…