Related papers: On computing factors of cyclotomic polynomials
Given f in Z[x_1,...,x_n], we compute the density of x in Z^n such that f(x) is squarefree, assuming the abc conjecture. Given f,g in Z[x_1,...,x_n], we compute unconditionally the density of x in Z^n such that gcd(f(x),g(x))=1. Function…
We consider the problem of efficiently computing isolated coefficients $c_n$ in the Fourier series of the elliptic modular function $j(\tau)$. We show that a hybrid numerical-modular method with complexity $n^{1+o(1)}$ is efficient in…
We pose the question of what is the best generalization of the factorial and the binomial coefficient. We give several examples, derive their combinatorial properties, and demonstrate their interrelationships. On cherche ici \`a…
We consider multiple orthogonal polynomials associated with the exponential cubic weight e^{-x^3} over two contours in the complex plane. We study the basic properties of these polynomials, including the Rodrigues formula and…
In this article we compute the $q$th power values of the quadratic polynomials $f$ with negative squarefree discriminant such that $q$ is coprime to the class number of the splitting field of $f$ over $\mathbb{Q}$. The theory of unique…
We present a new algorithm to decompose generic spinor polynomials into linear factors. Spinor polynomials are certain polynomials with coefficients in the geometric algebra of dimension three that parametrize rational conformal motions.…
The problem of determining cyclotomic numbers in terms of the solutions of certain Diophantine systems has been treated by many authors since the age of Gauss. In this paper we obtain an explicit expression for cyclotomic numbers of order…
We present a new probabilistic algorithm to compute modular polynomials modulo a prime. Modular polynomials parameterize pairs of isogenous elliptic curves and are useful in many aspects of computational number theory and cryptography. Our…
Taking a combinatorial point of view on cyclotomic polynomials leads to a larger class of polynomials we shall call the inclusion-exclusion polynomials. This gives a more appropriate setting for certain types of questions about the…
Let $R=K[x_{1},x_{2},\cdots, x_{m}]$ where $K$ is a field. In this paper, we give some properties of $n$-matrix factorizations of polynomials in $R$. We also derive some results giving some lower bounds on the number of $n$-matrix factors…
The main object of this paper is to find closed form expressions for finite and infinite sums that are weighted by $\omega(n)$, where $\omega(n)$ is the number of distinct prime factors of $n$. We then derive general convergence criteria…
In this article we use a method of finding the index of a complex-valued function by determined number of arithmetic operations to describe an algorithm of localization of roots of square-free polynomials. We give an estimation of the…
We prove four identities for the squared central binomial coefficients. The first three of them reflect certain transformation properties of the complete elliptic integrals of the first and the second kind, while the last one is based on…
We consider certificates of positivity for univariate polynomials with rational coefficients that are positive over (an interval of)~$\mathbb{R}$. Such certificates take the form of weighted sums of squares (SOS) of polynomials with…
We adapt the Faddeev-LeVerrier algorithm for the computation of characteristic polynomials to the computation of the Pfaffian of a skew-symmetric matrix. This yields a very simple, easy to implement and parallelize algorithm of…
A method of determining two factors of an odd integer without need of multiplication or division operation in iterative portion of computation is presented. It is feasible for an implementing algorithm to use only integer addition and…
We consider the polynomial equation $$X^n + a_{n-1}\cdot X^{n-1} + \dots + a_1 \cdot X + a_0 \cdot I = O,$$ over $(2 \times 2)$-matrices $X$ with the real entries, where $I$ is the identity matrix, $O$ is the null matrix, $a_i \in \mathbb…
After revisiting Cantor-Zassenhaus polynomial factorization algorithm, we describe a new simplified version of it, which requires less computational cost. Moreover we show that it is able to find a factor of a fully splitting polynomial of…
In this paper we develop techniques that eliminate the need of the Generalized Riemann Hypothesis (GRH) from various (almost all) known results about deterministic polynomial factoring over finite fields. Our main result shows that given a…
Let $f(x_1,...,x_k)$ be a polynomial over a field $K$. This paper considers such questions as the enumeration of the number of nonzero coefficients of $f$ or of the number of coefficients equal to $\alpha\in K^*$. For instance, if $K=\ff_q$…