Related papers: Some new canonical forms for polynomials

Invariants of generalized tensor fields on a line are classified using special polynomials P_mk^(-1/lambda) introduced here for this purpose. For the case of positive characteristic, a new invariant of formal power series, a width, is…

We use recent results on algorithms for Markov decision problems to show that a canonical form for a generalized P-matrix can be computed, in some important cases, by a strongly polynomial algorithm.

In this paper we use the theory of modular forms to find formulas for the number of representations of a positive integer by certain class of quadratic forms in eight variables, viz., forms of the form $a_1x_1^2 + a_2 x_2^2 + a_3 x_3^2 +…

Canonical matrices of (a) bilinear and sesquilinear forms, (b) pairs of forms, in which every form is symmetric or skew-symmetric, and (c) pairs of Hermitian forms are given over finite fields of characteristic not 2 and over finite…

We introduce techniques to analyze unitary operations in terms of quadratic form expansions, a form similar to a sum over paths in the computational basis when the phase contributed by each path is described by a quadratic form over…

This material is a rewriting and expansion of notes for beginning graduate students in seminars in combinatorics (Department of Mathematics, University of California San Diego). Solid skills in linear and multilinear algebra were required…

For every positive integer k, it is shown that there exists a positive definite diagonal quaternary integral quadratic form that represents all positive integers except for precisely those which lie in k arithmetic progressions. For k=1,…

We construct the quaternion algebra [10] "geometrically" by a three dimensional analogue of the classic two dimensional geometric description of the complex field. The algebraic description of the multiplication operation in three…

In this paper, a new construction of quaternary bent functions from quaternary quadratic forms over Galois rings of characteristic 4 is proposed. Based on this construction, several new classes of quaternary bent functions are obtained, and…

An explicit formula for the canonical bilinear form on the Grothendieck ring of the Lie supergroup $GL(n,m)$ is given. As an application we get an algorithm for the decomposition Euler characters in terms of characters of irreducible…

An observation by J-P. Serre implies that cubic polynomials are unique among generic monic polynomials of degree 2 or higher in that they have a root that is a power series in the discriminant of the polynomial. We provide formulas for this…

This paper examines with elementary proofs some interesting properties of numbers in the binary quadratic form $a^2+ab+b^2$, where $a$ and $b$ are non-negative integers. Key findings of this paper are (i) a prime number $p$ can be…

In this paper, we prove that a binary definite quadratic form over F_q[t], where q is odd, is completely determined up to equivalence by the polynomials it represents up to degree 3m-2, where m is the degree of its discriminant. We also…

We investigate exponential sums over singular binary quartic forms, proving an explicit formula for the finite field Fourier transform of this set. Our formula shares much in common with analogous formulas proved previously for other vector…

Lagrange's four-square theorem asserts that any $n\in\mathbb N=\{0,1,2,\ldots\}$ can be written as the sum of four squares. This can be further refined in various ways. We show that any $n\in\mathbb N$ can be written as $x^2+y^2+z^2+w^2$…

A positive integer $n$ is said to be a practical number if every integer in $[1,n]$ can be represented as the sum of distinct divisors of $n$. In this article, we consider practical numbers of a given polynomial form. We give a necessary…

For any integer $x$, let $T_x$ denote the triangular number $\frac{x(x+1)}{2}$. In this paper we give a complete characterization of all the triples of positive integers $(\alpha, \beta, \gamma)$ for which the ternary sums $\alpha x^2…

In this paper we generalize the result of Fouvry and Iwaniec dealing with prime values of the quadratic form $x^2 + y^2$ with one input restricted to a thin subset of the integers. We prove the same result with an arbitrary primitive…

A systematic study of non-trivial cubic extensions of the four-dimensional Poincar\'e algebra is undertaken. Explicit examples are given with various techniques (Young tableau, characters etc).

Projectivizations of pointed polyhedral cones $C$ are positive geometries in the sense of Arkani-Hamed, Bai, and Lam. Their canonical forms look like $$ \Omega_C(x)=\frac{A(x)}{B(x)} dx, $$ with $A,B$ polynomials. The denominator $B(x)$ is…