Related papers: The Waring loci of ternary quartics

The aim of this paper is twofold. Firstly, we investigate a finite sum involving the generalized falling factorial polynomials, in some special cases of which we express it in terms of the degenerate Stirling numbers of the second kind, the…

Let $f(x)=x^6+Ax^{2k}+B\in {\mathbb Z}[x]$, with $A\ne 0$ and $k\in \{1,2\}$. We say that $f(x)$ is {\em monogenic} if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,\theta,\theta^2,\theta^3,\theta^4,\theta^{5}\}$ is a basis for the ring…

We consider polynomials with integer coefficients and discuss their factorization properties in Z[[x]], the ring of formal power series over Z. We treat polynomials of arbitrary degree and give sufficient conditions for their reducibility…

Apolarity is an important tool in commutative algebra and algebraic geometry which studies a form, $f$, by the action of polynomial differential operators on $f$. The quotient of all polynomial differential operators by those which…

We say that a monic polynomial $f(x)\in {\mathbb Z}[x]$ of degree $N$ is monogenic if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,\theta,\theta^2,\ldots ,\theta^{N-1}\}$ is a basis for ${\mathbb Z}_K$, the ring of integers of…

Let $n\ge1$, $r\ge0$ and $s\ge0$ be integers satisfying $4+r+3 s\le3^{n+1}$. Given linear polynomials $f_{i}(x)=m_{i} x+n_{i}$ for $1 \le i \le r+s$, where the coefficients $m_{i} , n_{i}$ are positive integers satisfying certain…

Let $G(k)$ denote the least number $s$ having the property that every sufficiently large natural number is the sum of at most $s$ positive integral $k$-th powers. Then for all $k\in \mathbb N$, one has \[ G(k)\le \lceil k(\log…

We investigate factorizability of a quadratic split quaternion polynomial. In addition to inequality conditions for existence of such factorization, we provide lucid geometric interpretations in the projective space over the split…

Let $f$ be a (possibly Newton degenerate) weighted homogeneous polynomial defining an isolated surface singularity at the origin of $\mathbb{C}^3$, and let $\{f_s\}$ be a generic deformation of its coefficients such that $f_s$ is Newton…

I investigate on the number t of real eigenvectors of a real symmetric tensor. In particular, given a homogeneous polynomial f of degree d in 3 variables, i prove that t is greater or equal than 2c+1, if d is odd and t is greater or equal…

Let $ p \ge 5 $ be a prime and let $ b, c \in \mathbb{Z} $. Denote by $ T_k(b,c) $ the generalized central trinomial coefficient, i.e., the coefficient of $ x^k $ in $ (x^2 + bx + c)^k $. In this paper, we establish congruences modulo $ p^3…

While every matrix admits a singular value decomposition, in which the terms are pairwise orthogonal in a strong sense, higher-order tensors typically do not admit such an orthogonal decomposition. Those that do have attracted attention…

We study in detail the zero set of a regular function of a quaternionic or octonionic variable. By means of a division lemma for convergent power series, we find the exact relation existing between the zeros of two octonionic regular…

We consider representations of real forms of even degree as a linear combination of powers of real linear forms, counting the number of positive and negative coefficients. We show that the natural generalization of Sylvester's Law of…

We prove that for all integers $k \geq 1$, $q\ge (k-1)^4+ 6k$, and $m \geq 1$, every matrix in $ M_m(\mathbb F_q)$ is a sum of two kth powers: $M_m(\mathbb F_q)=\{A^k+B^k|A,B\in M_m(\mathbb F_q)\}$. We further generalize and refine this…

A monic polynomial $f(x)\in {\mathbb Z}[x]$ of degree $N$ is called monogenic if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,\theta,\theta^2,\ldots ,\theta^{N-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(\theta)$, where…

The present article is concerned with global subelliptic estimates for Kramers-Fokker-Planck operators with polynomials of degree less than or equal to two. The constants appearing in those estimates are accurately formulated in terms of…

The simultaneous invariants of 2, 3, 4 and 5 ternary quadratic forms under the group $\SL(3, {\Bbb C})$ were given by several authors (P. Gordan, C. Ciamberlini, H.W. Turnbull, J.A Todd), utilizing the symbolic method. Using the Jordan…

Consider a quartic number field $K$ generated by a root of an irreducible quadrinomial of the form $ F(x)= x^4+ax^3+bx+c \in \Z[x]$. Let $i(K)$ denote the index of $K$. Engstrom \cite{Engstrom} established that $i(K)=2^u \cdot 3^v$ with $u…

Many questions in number theory concern the nonvanishing of determinants of square matrices of logarithms (complex or p-adic) of algebraic numbers. We present a new conjecture that states that if such a matrix has vanishing determinant,…