Related papers: Quotient polynomials with positive coefficients
Let F be a homogeneous real polynomial of even degree in any number of variables. We consider the problem of giving explicit conditions on the coefficients so that F is positive definite or positive semi-definite. In this note we produce a…
Linear differential equations of arbitrary order with polynomial coefficients are considered. Specifically, necessary and sufficient conditions for the existence of polynomial solutions of a given degree are obtained for these equations. An…
We present a necessary and sufficient condition for a cubic polynomial to be positive for all positive reals. We identify the set where the cubic polynomial is nonnegative but not all positive for all positive reals, and explicitly give the…
We establish necessary and sufficient conditions for a polynomial to be divisible by a cyclotomic polynomials and derive new formulas involving Ramanujan sums as an application of our results. Additionally, we provide new insights into the…
In this article, we prove some factorization results for several classes of polynomials having integer coefficients, which in particular yield several classes of irreducible polynomials. Such classes of polynomials are devised by imposing…
We give necessary and sufficient existence criteria, and methods for finding, continuous solutions of linear equations whose coefficients are polynomials.
We show that if two monic polynomials with integer coefficients have square-free resultant, then all positive divisors of the resultant arise as the greatest common divisor of the values of the two polynomials at a suitable integer.
When the Euclidean algorithm produces a symmetric sequence of quotients, we give explicit formulas for the remainders that allow the analysis of two families of quadratic forms in the remainders.
How to handle division in systems that compute with logical formulas involving what would otherwise be polynomial constraints over the real numbers is a surprisingly difficult question. This paper argues that existing approaches from both…
We describe the limit zero distributions of sequences of polynomials with positive coefficients.
We give necessary conditions satisfied by the set of odd prime divisors of binary perfect polynomials. This allows us to get a new characterization of all the known perfect binary polynomials.
In this paper, we give a formula for the proper class number of a binary quadratic polynomial assuming that the conductor ideal is sufficiently divisible at dyadic places. This allows us to study the growth of the proper class numbers of…
A recently-established necessary condition for polynomials that preserve the class of entrywise nonnegative matrices of a fixed order is shown to be necessary and sufficient for the class of nonnegative monomial matrices. Along the way, we…
Using polynomial evaluation, we give some useful criteria to answer questions about divisibility of polynomials. This allows us to develop interesting results concerning the prime elements in the domain of coefficients. In particular, it is…
In this paper, we prove several theorems on systems of polynomials with at least one positive real zero based on the theory of conceive polynomials. These theorems provide sufficient conditions for systems of multivariate polynomials…
We describe efficient algorithms to search for cases in which binomial coefficients are equal or almost equal, give a conjecturally complete list of all cases where two binomial coefficients differ by 1, and give some identities for…
We give necessary and sufficient conditions for existence and infinite divisibility of $\alpha$-determinantal processes. For that purpose we use results on negative binomial and ordinary binomial multivariate distributions.
We formulate and prove a necessary condition for a sequence of analytic trigonometric polynomials with real non-negative coefficients to be flat a.e.
We introduce the concept of quotient in PN spaces and give some examples. We prove some theorems with regard to the completeness of a quotient.
We classify the polynomials with integral coefficients that, when evaluated on a group element of finite order $n$, define a unit in the integral group ring for infinitely many positive integers $n$. We show that this happens if and only if…