On some classes of polynomials with nonnegative coefficients and a given factor

For a given real polynomial f without positive roots we study polynomials g of lowest degree such that the product gf has positive (nonnegative, respectively) coefficients. We show that for quadratic f with negative linear coefficient every such g must have positive coefficients and exhibit an easy procedure for the determination of g. If f has only integer coefficients we show that g with integer coefficients can be found. Furthermore, for some classes of polynomials f we give upper (lower, respectively) bounds for the degrees of g.     

Linear relations among algebraic solutions of differential equations

For any univariate polynomial with coefficients in a differential field of characteristic zero and any integer, q, there exists an associated nonzero linear ordinary differential equation (LODE) with the following two properties. Each term of the LODE lies in the differential field generated by the rational numbers and the coefficients of the polynomial, and the qth power of each root of the polynomial is a solution of this LODE. This LODE is called a qth power resolvent of the polynomial. We will show how one can get a resolvent for the logarithmic derivative of the roots of a polynomial from the αth power resolvent of the polynomial, where α is an indeterminate that takes the place of q. We will demonstrate some simple relations among the algebraic and differential equations for the roots and their logarithmic derivatives. We will also prove several theorems regarding linear relations of roots of a polynomial over constants or the coefficient field of the polynomial depending upon the (nondifferential) Galois group. Finally, we will use a differential resolvent to solve the Riccati equation.     

Polynomial root radius optimization with affine constraints

The root radius of a polynomial is the maximum of the moduli of its roots (zeros). We consider the following optimization problem: minimize the root radius over monic polynomials of degree n, with either real or complex coefficients, subject to k linearly independent affine constraints on the coefficients. We show that there always exists an optimal polynomial with at most \(k-1\) inactive roots, that is, roots whose moduli are strictly less than the optimal root radius. We illustrate our results using some examples arising in feedback control.     

On realizability of sign patterns by real polynomials

The classical Descartes’ rule of signs limits the number of positive roots of a real polynomial in one variable by the number of sign changes in the sequence of its coefficients. One can ask the question which pairs of nonnegative integers (p, n), chosen in accordance with this rule and with some other natural conditions, can be the pairs of numbers of positive and negative roots of a real polynomial with prescribed signs of the coefficients. The paper solves this problem for degree 8 polynomials.     

Polynomials with nonnegative coefficients and a given factor

For a given real polynomial f without positive roots (without nonnegative roots, respectively) we study polynomials h of lowest degree such that the product hf has nonnegative (positive, respectively) coefficients. Our main result is the determination of polynomials h of minimal degree with the aforementioned properties for quadratic polynomials f with negative discriminant; here we exploit slightly modified ideas of E. Meissner (1911). Further, we show that h can be calculated in finitely many steps.     

Inverse cyclotomic polynomials

Let Ψn(x) be the monic polynomial having precisely all non-primitive nth roots of unity as its simple zeros. One has Ψn(x)=(xn−1)/Φn(x), with Φn(x) the nth cyclotomic polynomial. The coefficients of Ψn(x) are integers that like the coefficients of Φn(x) tend to be surprisingly small in absolute value, e.g. for n<561 all coefficients of Ψn(x) are ?1 in absolute value. We establish various properties of the coefficients of Ψn(x), especially focusing on the easiest non-trivial case where n is composed of 3 distinct odd primes.     

Schanuel's Conjecture and Algebraic Roots of Exponential Polynomials

In this article, I will prove that assuming Schanuel's conjecture, an exponential polynomial with algebraic coefficients can have only finitely many algebraic roots. Furthermore, this proof demonstrates that there are no unexpected algebraic roots of any such exponential polynomial. This implies a special case of Shapiro's conjecture: if p(x) and q(x) are two exponential polynomials with algebraic coefficients, each involving only one iteration of the exponential map, and they have common factors only of the form exp (g) for some exponential polynomial g, then p and q have only finitely many common zeros.     

On cubic Galois field extensions

We study Morton's characterization of cubic Galois extensions F/K by a generic polynomial depending on a single parameter s∈K. We show how such an s can be calculated with the coefficients of an arbitrary cubic polynomial over K the roots of which generate F. For K=Q we classify the parameters s and cubic Galois polynomials over Z, respectively, according to the discriminant of the extension field, and we present a simple criterion to decide if two fields given by two s-parameters or defining polynomials are equal or not.     

On arrangements of roots for a real hyperbolic polynomial and its derivatives

In this paper we count the number ?_n^(0,k), k?n−1, of connected components in the space Δ_n^(0,k) of all real degree n polynomials which a) have all their roots real and simple; and b) have no common root with their kth derivatives. In this case, we show that the only restriction on the arrangement of the roots of such a polynomial together with the roots of its kth derivative comes from the standard Rolle's theorem. On the other hand, we pose the general question of counting all possible root arrangements for a polynomial p(x) together with all its nonvanishing derivatives under the assumption that the roots of p(x) are real. Already the first nontrivial case n=4 shows that the obvious restrictions coming from the standard Rolle's theorem are insufficient. We prove a generalized Rolle's theorem which gives an additional restriction on root arrangements for polynomials.     

Bezoutians and projections

With each polynomial p of degree n whose roots lie inside the unit disc we may associate the n-dimensional space of all solutions of the recurrence relation whose coefficients are those of p (considered as a subspace of 1²). The main result consists in establishing a close relation between the Bezoutian of two such polynomials (of the same degree) and the projection operator onto one of the corresponding spaces along the complement of the other. The note forms a loose continuation of the author's investigations of the infinite companion matrix—the generating function of the infinite companion matrix of a polynomial p appears thus as a particular case; the corresponding Bezoutian is that of the pair p and zⁿ.     

Attracting cycles for the relaxed Newton’s method

We study the relaxed Newton’s method applied to polynomials. In particular, we give a technique such that for any n≥2, we may construct a polynomial so that when the method is applied to a polynomial, the resulting rational function has an attracting cycle of period n. We show that when we use the method to extract radicals, the set consisting of the points at which the method fails to converge to the roots of the polynomial p(z)=z^m−c (this set includes the Julia set) has zero Lebesgue measure. Consequently, iterate sequences under the relaxed Newton’s method converge to the roots of the preceding polynomial with probability one.     

Roots and polynomials as Homeomorphic spaces

We provide a unified, elementary, topological approach to the classical results stating the continuity of the complex roots of a polynomial with respect to its coefficients, and the continuity of the coefficients with respect to the roots. In fact, endowing the space of monic polynomials of a fixed degree n and the space of n roots with suitable topologies, we are able to formulate the classical theorems in the form of a homeomorphism. Related topological facts are also considered.     

On center sets of ODEs determined by moments of their coefficients

The classical H. Poincaré Center-Focus problem asks about the characterization of planar polynomial vector fields such that all their integral trajectories are closed curves whose interiors contain a fixed point, a center. This problem can be reduced to a center problem for some ordinary differential equation whose coefficients are trigonometric polynomials depending polynomially on the coefficients of the field. In this paper we show that the set of centers in the Center-Focus problem can be determined as the set of zeros of some continuous functions from the moments of coefficients of this equation.     

Root statistics of random polynomials with bounded Mahler measure

The Mahler measure of a polynomial is a measure of complexity formed by taking the modulus of the leading coefficient times the modulus of the product of its roots outside the unit circle. The roots of a real degree N polynomial chosen uniformly from the set of polynomials of Mahler measure at most 1 yield a Pfaffian point process on the complex plane. When N is large, with probability tending to 1, the roots tend to the unit circle, and we investigate the asymptotics of the scaled kernel in a neighborhood of a point on the unit circle. When this point is away from the real axis (on which there is a positive probability of finding a root) the scaled process degenerates to a determinantal point process with the same local statistics (i.e.   scalar kernel) as the limiting process formed from the roots of complex polynomials chosen uniformly from the set of polynomials of Mahler measure at most 1. Three new matrix kernels appear in a neighborhood of ±1 which encode information about the correlations between real roots, between complex roots and between real and complex roots. Away from the unit circle, the kernels converge to new limiting kernels, which imply among other things that the expected number of roots in any open subset of C$\mathbb{C}$ disjoint from the unit circle converges to a positive number. We also give ensembles with identical statistics drawn from two-dimensional electrostatics with potential theoretic weights, and normal matrices chosen with regard to their topological entropy as actions on Euclidean space.     

Explicit determination of generalized symmetric and alternating Galois groups

This paper shows a probabilistic algorithm to decide whether the Galois group of a given irreducible polynomial with rational coefficients is the generalized symmetric group C_p?S_m or the generalized alternating group C_p?A_m. In the affirmative case, we give generators of the group with their action on the set of roots of the polynomial.     

On symmetries of roots of rational functions and the classification of rational function solutions of functional equations arising from multiplication of quantum integers with prime semigroup supports

The study of quantum integers and their operations is closely related to the studies of symmetries of roots of polynomials and of fundamental questions of decompositions in Additive Number Theory. In his papers on quantum arithmetics, Melvyn Nathanson raises the question of classifying solutions of functional equations arising from the multiplication of quantum integers, starting with polynomial solutions and then generalizing to rational function solutions. The classification of polynomial solutions with fields of coefficients of characteristic zero and support base P has been completed. In a paper concerning the Grothendieck group associated to the collection of polynomial solutions, Nathanson poses a problem which asks whether the set of rational function solutions strictly contains the set of ratios of polynomial solutions. It is now known that there are infinitely many rational function solutions \(\Gamma \) with fields of coefficients of characteristic zero not constructible as ratios of polynomial solutions, even in the purely cyclotomic case, which is the case most similar to the polynomial solution case. The classification of polynomial solutions is thus not sufficient, in essential ways, to resolve the classification problem of all rational function solutions with fields of coefficients of characteristic zero. In this paper we study symmetries of roots of rational functions and the classification of the important class-the last and main obstruction to the classification problem-of rational function solutions, the purely cyclotomic, purely nonrational primitive solutions with fields of coefficients of characteristic zero and support base P, which allows us to complete the classification problem raised by Nathanson.     

New classes of spectrally arbitrary ray patterns

An n×n ray pattern matrix S is said to be spectrally arbitrary if for every monic nth degree polynomial f(λ) with coefficients from C, there is a complex matrix in the ray pattern class of S such that its characteristic polynomial is f(λ). In this article we give new classes of spectrally arbitrary ray pattern matrices.     

Eigenvalue inequalities for principal submatrices

For a polynomial with real roots, inequalities between those roots and the roots of the derivative are demonstrated and translated into eigenvalue inequalities for a hermitian matrix and its submatrices. For example, given an n-by-n positive definite hermitian matrix with maximum eigenvalue λ, these inequalities imply that some principal submatrix has an eigenvalue exceeding $\text{[(}\text{n?1)}\text{n}\text{]λ}$.     

Complete solution of the polynomial version of a problem of Diophantus

In this paper, we prove that if {a,b,c,d} is a set of four non-zero polynomials with integer coefficients, not all constant, such that the product of any two of its distinct elements plus 1 is a square of a polynomial with integer coefficients, then

(a+b−c−d)2=4(ab+1)(cd+1).     

The recovery of even polynomial potentials

We consider the problem of reconstructing an even polynomial potential from one set of spectral data of a Sturm-Liouville problem. We show that we can recover an even polynomial of degree 2m from m+1 given Taylor coefficients of the characteristic function whose zeros are the eigenvalues of one spectrum. The idea here is to represent the solution as a power series and identify the unknown coefficients from the characteristic function. We then compute these coefficients by solving a nonlinear algebraic system, and provide numerical examples at the end. Because of its algebraic nature, the method applies also to non self-adjoint problems.