Related papers: Quotient polynomials with positive coefficients

We obtain explicit upper bounds for the number of irreducible factors for a class of compositions of polynomials in several variables over a given field. In particular, some irreducibility criteria are given for this class of compositions…

For a fixed quadratic irreducible polynomial $f$ with no fixed prime factors at prime arguments, we prove that there exist infinitely many primes $p$ such that $f(p)$ has at most 4 prime factors, improving a classical result of Richert who…

In this note we give a theoretical support by means of quotient polynomial rings for the computation formulas of the dimension of abelian codes.

Polynomial approximations of functions are widely used in scientific computing. In certain applications, it is often desired to require the polynomial approximation to be non-negative (resp. non-positive), or bounded within a given range,…

We determine necessary conditions for when powers corresponding to positive/negative coefficients of $\Phi_{n}$ are in arithmetic progression. When $n = pq$ for any primes $q>p>2$, our conditions are also sufficient. Finally, we generalize…

In this article, we give an account of some recent irreducibility testing criteria for polynomials having integer coefficients over the field of rational numbers.

We give an algorithm determining whether a hermiticity-preserving superoperator is positive. In our approach we apply techniques of quantifier elimination theory for real numbers. Furthermore, we argue that quantifier elimination theory…

Real algebraic geometry provides certificates for the positivity of polynomials on semi-algebraic sets by expressing them as a suitable combination of sums of squares and the defining inequalitites. We show how Putinar's theorem for…

Two polynomials, $f,g \in \mathbb{Z}[x]$ are evaluationally coprime at x if $\gcd(f(x),g(x))=1$. We give necessary and sufficient conditions for two such linear polynomials to have a positive proportion of evaluated coprime values.

We investigate polynomials, called m-polynomials, whose generator polynomial has coefficients that can be arranged as a matrix, where q is a positive integer greater than one. Orthogonality relations are established and coefficients are…

We study properties of a category after quotienting out a suitable chosen group of isomorphisms on each object. Coproducts in the original category are described in its quotient by our new weaker notion of a 'phased coproduct'. We examine…

In this paper, we derive some formulae involving coefficients of polynomials which occur quite naturally in the study of restricted partitions. Our method involves a recently discovered sieve technique by Li and Wan (Sci. China. Math.…

In this paper one extends the binomial and trinomial coefficients to the concept of 'k-nomial' coefficients, and one obtains some properties of these. As an application one generalizes Pascal's triangle.

Least-squares optimized polynomials are discussed which are needed in the two-step multi-bosonic algorithm for Monte Carlo simulations of quantum field theories with fermions. A recurrence scheme for the calculation of necessary…

Gradient-based algorithms, popular strategies to optimization problems, are essential for many modern machine-learning techniques. Theoretically, extreme points of certain cost functions can be found iteratively along the directions of the…

A partition polynomial is a refinement of the partition number p(n) whose coefficients count some special partition statistic. Just as partition numbers have useful asymptotics so do partition polynomials. In fact, their asymptotics…

We consider the numerical integration of non-autonomous separable parabolic equations using high order splitting methods with complex coefficients (methods with real coefficients of order greater than two necessarily have negative…

We use $q$-binomial theorem to prove three new polynomial identities involving $q$-trinomial coefficients. We then use summation formulas for the $q$-trinomial coefficients to convert our identities into another set of three polynomial…

We show that linear differential operators with polynomial coefficients over a field of characteristic zero can be multiplied in quasi-optimal time. This answers an open question raised by van der Hoeven.

We show that the proportion of polynomials of degree $n$ over the finite field with $q$ elements, which have a divisor of every degree below $n$, is given by $c_q n^{-1} + O(n^{-2})$. More generally, we give an asymptotic formula for the…