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We establish a necessary and sufficient condition for all zeros of a self-reciprocal polynomial to lie on the unit circle. Moreover, we relate the necessary and sufficient condition with a canonical system of linear differential equations…

In this note, we provide explicit expressions for the projections onto the graph of a quadratic polynomial. The projections are obtained by examining the critical points of the associated quartic polynomial, that is, the roots of the cubic…

Let X be a real nondegenerate projective subvariety such that its set of real points is Zariski dense. We prove that every real quadratic form that is nonnegative on X is a sum of squares of linear forms if and only if X is a variety of…

We show that every monic polynomial of degree three with complex coefficients and no repeated roots is either a (vertical and horizontal) translation of $y=x^3$ or can be composed with a linear function to obtain a Ramanujan cubic. As a…

For any real numbers $a,\ b$, and $c$, we form the sequence of polynomials $\{P_n(z)\}_{n=0}^\infty$ satisfying the four-term recurrence \[ P_n(z)+azP_{n-1}(z)+bP_{n-2}(z)+czP_{n-3}(z)=0,\ n\in\mathbb{N}, \] with the initial conditions…

In this article, we consider the polynomials of the form $f(x)=a_0+a_1x+a_2x^2+\cdots+a_nx^n\in \mathbb{Z}[x],$ where $|a_0|=|a_1|+\dots+|a_n|$ and $|a_0|$ is a prime. We show that these polynomials have a cyclotomic factor whenever…

We describe all special curves in the parameter space of complex cubic polynomials, that is all algebraic irreducible curves containing infinitely many post-critically finite polynomials. This solves in a strong form a conjecture by Baker…

This article provides a simple trigonometric method for determining how many roots of a quartic equation are real and how many are complex, without solving the equation. The approach replaces the quartic's classical discriminant -- a…

For a set $S$ of quadratic polynomials over a finite field, let $C$ be the (infinite) set of arbitrary compositions of elements in $S$. In this paper we show that there are examples with arbitrarily large $S$ such that every polynomial in…

We show that all $p$-adic quintic forms in at least $n>4562911$ variables have a non-trivial zero. We also derive new result concerning systems of cubic and quadratic forms.

We prove that the Jacobian conjecture is false if and only if there exists a solution to a certain system of polynomial equations. We analyse the solution set of this system. In particular we prove that it is zero dimensional.

In this paper, we are concerned with the problem of locating the zeros of polynomials of a quaternionic variable with quaternionic coefficients. We derive some new Cauchy bounds for the zeros of a polynomial by virtue of maximum modulus…

A class of self-inversive polynomials includes all the self-reciprocal polynomials. Let A denote the set of all self-reciprocal polynomials with n+1 coefficients. Let B denote the set of certain self-inversive and non self-reciprocal…

We estimate the number of primes represented by a general quadratic polynomial with discriminant $\Delta$, assuming that the corresponding real character is exceptional.

We give an optimal necessary and sufficient condition for the quotient polynomial and remainder in the division algorithm to have positive coefficients.

Let $f$ be an ordinary polynomial in $\mathbb{C}[z_1,..., z_n]$ with no negative exponents and with no factor of the form $z_1^{\alpha_1}... z_n^{\alpha_n}$ where $\alpha_i$ are non zero natural integer. If we assume in addicting that $f$…

Every real hyperbolic form in three variables can be realized as the determinant of a linear net of Hermitian matrices containing a positive definite matrix. Such representations are an algebraic certificate for the hyperbolicity of the…

We present an algorithm to compute all factorizations into linear factors of univariate polynomials over the split quaternions, provided such a factorization exists. Failure of the algorithm is equivalent to non-factorizability for which we…

A bivariate quartic form is a homogeneous bivariate polynomial of degree four. A criterion of positivity for such a form is known. In the present paper this criterion is reformulated in terms of pseudotensorial invariants of the form.

We show that if a real trigonometric polynomial has few real roots, then the trigonometric polynomial obtained by writing the coefficients in reverse order must have many real roots. This is used to show that a class of random trigonometric…