Monic Polynomials Research Articles

On monogenity of certain number fields defined by x 2·3 s ·q r − m

Let K = Q ( α ) , where α is a root of the monic irreducible polynomial x 2 · 3 s · q r − m with m ≠ ± 1 is a square-free integer, r and s are two positive integers, and q is a prime of the form 3 k + 2 . In this article, we study the monogenity of the number field K and establish conditions on m for K to be monogenic. Further, we provide sufficient conditions on r, s and m for K to be not monogenic. Finally, we illustrate our results with examples.

Microstrip Lowpass Filter Realization Using Monic Chebyshev Polynomial

This paper presents a novel approach to microwave lowpass filter design that addresses several challenges in modern microwave systems, including size, performance, and cost. With that in mind, we propose using monic Chebyshev polynomials as the characteristic function for the filter to achieve a low passband insertion loss and maximum cutoff slope. The half-power cutoff frequency characterizes these filters, and the ripple factor is not considered a design parameter. To demonstrate the effectiveness of the monic Chebyshev approximation, a comparison is made with chained function (CF) filters designed for microwave applications. It has been shown that the monic Chebyshev filter outperforms CF filters in all applications, including microwaves.

On the monogenity and Galois group of certain classes of polynomials

Abstract We say a monic polynomial g(x) ∈ ℤ[x] of degree n is monogenic if g(x) is irreducible over ℚ and {1, θ, …, θ n−1} is a basis for the ring ℤ K of integers of number field K = ℚ(θ), where θ is a root of g(x). Let f ( x ) = x n + c ∑ i = 1 n ( a x ) n − i ∈ Z [ x ] and F ( x ) = x n + c ∑ i = 1 n a i − 1 x n − i ∈ Z [ x ] $\begin{array}{} \displaystyle f(x)=x^n+c\sum_{i=1}^{n}(ax)^{n-i} \in \mathbb{Z}[x] \,\text{and}\, F(x)=x^n+c\sum_{i=1}^{n}a^{i-1}x^{n-i} \in \mathbb{Z}[x] \end{array}$ be irreducible polynomials having degree n ≥ 3. In this paper, we provide necessary and sufficient conditions involving only a, c, n for the polynomials f(x) and F(x) to be monogenic. As an application, we also provide a class of polynomials having a non square-free discriminant and Galois group Sn , the symmetric group on n letters.

MONOGENIC QUARTIC POLYNOMIALS AND THEIR GALOIS GROUPS

Abstract A monic polynomial $f(x)\in {\mathbb Z}[x]$ of degree N is called monogenic if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,\theta ,\theta ^2,\ldots ,\theta ^{N-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(\theta )$ , where $f(\theta )=0$ . We use the classification of the Galois groups of quartic polynomials, due to Kappe and Warren [‘An elementary test for the Galois group of a quartic polynomial’, Amer. Math. Monthly96(2) (1989), 133–137], to investigate the existence of infinite collections of monogenic quartic polynomials having a prescribed Galois group, such that each member of the collection generates a distinct quartic field. With the exception of the cyclic case, we provide such an infinite single-parameter collection for each possible Galois group. We believe these examples are new and we provide evidence to support this belief by showing that they are distinct from other infinite collections in the literature. Finally, we devote a separate section to the cyclic case.

Monodromy of stratified braid groups, II

The space of monic squarefree polynomials has a stratification according to the multiplicities of the critical points, called the equicritical stratification. Tracking the positions of roots and critical points, there is a map from the fundamental group of a stratum into a braid group. We give a complete determination of this map. It turns out to be characterized by the geometry of the translation surface structure on CP1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb{C}\\mathbb{P}^1$$\\end{document} induced by the logarithmic derivative df/f\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{d}f/f$$\\end{document} of a polynomial in the stratum.

On prescribed characteristic polynomials

Let F be a field. We show that given any nth degree monic polynomial q(x)∈F[x] and any matrix A∈Mn(F) whose trace coincides with the trace of q(x) and consisting in its main diagonal of k 0-blocks of order one, with k<n−k, and an invertible non-derogatory block of order n−k, we can construct a square-zero matrix N such that the characteristic polynomial of A+N is exactly q(x). We also show that the restriction k<n−k is necessary in the sense that, when the equality k=n−k holds, not every characteristic polynomial having the same trace as A can be obtained by adding a square-zero matrix. Finally, we apply our main result to decompose matrices into the sum of a square-zero matrix and some other matrix which is either diagonalizable, invertible, potent or torsion.

On 2-superirreducible polynomials over finite fields

We investigate k-superirreducible polynomials, by which we mean irreducible polynomials that remain irreducible under any polynomial substitution of positive degree at most k. Let F be a finite field of characteristic p. We show that no 2-superirreducible polynomials exist in F[t] when p=2 and that no such polynomials of odd degree exist when p is odd. We address the remaining case in which p is odd and the polynomials have even degree by giving an explicit formula for the number of monic 2-superirreducible polynomials having even degree d. This formula is analogous to that given by Gauss for the number of monic irreducible polynomials of given degree over a finite field. We discuss the associated asymptotic behaviour when either the degree of the polynomial or the size of the finite field tends to infinity.

Generalized Gauss-Rys orthogonal polynomials

Let (Pn(x;z;λ))n≥0 be the sequence of monic orthogonal polynomials with respect to the symmetric linear functional s defined by〈s,p〉=∫−11p(x)(1−x2)(λ−1/2)e−zx2dx,λ>−1/2,z>0. In this contribution, several properties of the polynomials Pn(x;z;λ) are studied taking into account the relation between the parameters of the three-term recurrence relation that they satisfy. Asymptotic expansions of these coefficients are given. Discrete Painlevé and Painlevé equations associated with such coefficients appear naturally. An electrostatic interpretation of the zeros of such polynomials as well as the dynamics of the zeros in terms of the parameters z and λ are given.

Interpolation by decomposable univariate polynomials

The usual univariate interpolation problem of finding a monic polynomial f of degree n that interpolates n given values is well understood. This paper studies a variant where f is required to be composite, say, a composition of two polynomials of degrees d and e, respectively, with de=n, and with d+e−1 given values. Some special cases are easy to solve, and for the general case, we construct a homotopy between it and a special case. We compute a geometric solution of the algebraic curve presenting this homotopy, and this also provides an answer to the interpolation task. The computing time is polynomial in the geometric data, like the degree, of this curve. A consequence is that for almost all inputs, a decomposable interpolation polynomial exists.

Cubic decomposition of a Laguerre–Hahn linear functional I

The aim of this contribution is to study orthogonal polynomials via cubic decomposition in the framework of the Laguerre–Hahn class. We consider two monic orthogonal polynomial sequences \{W_{n}\}_{n\geq0} and \{P_{n}\}_{n\geq0} , and we let w and u be, respectively, the corresponding regular linear functionals such that W_{3n}(x)=P_{n}(x^{3}) , n\geq0 . We prove that if either w or u is a Laguerre–Hahn linear functional, then so is the other one. Based on this result, we deduce a complete analysis of the class s of the Laguerre–Hahn linear functional w . More precisely, we show that 3s'\leq s\leq3s'+6 , where s' is the class of u . An illustrative example of class 1 is analyzed.

Number of components of polynomial lemniscates: A problem of Erdös, Herzog, and Piranian

Let K⊂C be a compact set in the plane whose logarithmic capacity c(K) is strictly positive. Let Pn(K) be the space of monic polynomials of degree n, all of whose zeros lie in K. For p∈Pn(K), its filled unit leminscate is defined by Λp={z:|p(z)|<1}. Let C(Λp) denote the number of connected components of the open set Λp, and define Cn(K)=maxp∈Pn(K)⁡C(Λp). In this paper we show that the quantityM(K)=limsupn→∞Cn(K)n, satisfies M(K)<1 when the logarithmic capacity c(K)<1, and M(K)=1 when c(K)≥1. In particular, this answers a question of Erdös et al. posed in (1958) [8]. In addition, we show that for nice enough compact sets whose capacity is strictly bigger than 12, the quantity m(K)=liminfn→∞Cn(K)n>0.

Monogenity of iterates of irreducible binomials

One of the oldest problems of algebraic number theory is to find a method to determine if the ring of integers of a number field K is Z [ θ ] for some θ ∈ K ; a field for which the answer to this question is affirmative is referred to as a monogenic field. Suppose f ( x ) = x m − a ∈ Z [ x ] is a monic irreducible polynomial and α n ∈ C is a root of f n ( x ) , the n-fold composition of f. In this article, we prove a necessary and sufficient condition for K n = Q ( α n ) to be monogenic for every n ∈ N and m ≥ 2.

On the ideal discriminant of some relative pure extensions

Let L = K(α) be an extension of a number field K where α satisfies the monic irreducible polynomial P(X) = Xp −a ∈ R[X] of prime degree p and such that a is pth power free in R := OK (the ring of integers of K). The purpose of this paper is to give an explicit formula for the ideal discriminant DL/K of L over K involving only the prime ideals dividing the principal ideals aR and pR. As an illustration, we compute the discriminant DL/K of a family of septic and quintic pure fields over quadratic fields. Hence a slightly simpler computation of discriminant DL/K is obtained.

The On power integral bases of certain pure number fields defined by $x^{120}-m $

Let $K$ be a pure number field with $\alpha$ a complex root of a monic irreducible polynomial $F(x) = x^{120}-m \in \mathbb{Z}[x]$ with $ m \neq \pm 1 $. In this paper, we study the monogenity of $K$. More precisely, we prove that if $m$ is square-free, $m \not \equiv 1\md{4}$, $m \not \equiv \pm 1 \md{9} $, and $\ol{m}\not \in \{ \mp 1, 7, 18 \} \md {25}$, then $K$ is monogenic. On the other hand, if $m \equiv 1\md{4}$, $m \equiv 1 \md{9} $, or $m \equiv 1 \md{25}$, then $K$ is not monogenic. Our results are illustrated by some computational examples.

Fractional parts of generalized polynomials at prime arguments

Let f(X) be a monic polynomial of degree d≥2 with real coefficients. In this paper, we obtain an upper bound for the discrepancy of the sequence ([f(p)α]β) generated by the generalized polynomial [f(X)α]β, where p runs through the set of all primes and α, β are irrational numbers satisfying certain conditions. We also study the small fractional parts of the sequence ([f(p)α]β) and prove a result analogous to that of Madritsch and Tichy (Theorem 2.3, [13]).

Equidistribution of high traces of random matrices over finite fields and cancellation in character sums of high conductor

AbstractLet be a random matrix distributed according to uniform probability measure on the finite general linear group . We show that equidistributes on as as long as and that this range is sharp. We also show that nontrivial linear combinations of equidistribute as long as and this range is sharp as well. Previously equidistribution of either a single trace or a linear combination of traces was only known for , where depends on , due to work of the first author and Rodgers. We reduce the problem to exhibiting cancellation in certain short character sums in function fields. For the equidistribution of , we end up showing that certain explicit character sums modulo exhibit cancellation when averaged over monic polynomials of degree in as long as . This goes far beyond the classical range due to Montgomery and Vaughan. To study these sums, we build on the argument of Montgomery and Vaughan but exploit additional symmetry present in the considered sums.

On common index divisors and monogenity of certain number fields defined by x^{5}+ax+b

Let K = Q(α) be a number field, where α satisfies the monic irreducible polynomial F (x) = x5 + ax + b belonging to Z[x]. The purpose of this paper is to caracterise when a prime p is a common index divisor of K. More precisely, we give explicitly a necessary and sufficient conditions, on a and b for which K is not monogenic. Some useful examples are also given.

Saddle point braids of braided fibrations and pseudo-fibrations

Let gt\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g_t$$\\end{document} be a loop in the space of monic complex polynomials in one variable of fixed degree n. If the roots of gt\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g_t$$\\end{document} are distinct for all t, they form a braid B1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_1$$\\end{document} on n strands. Likewise, if the critical points of gt\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g_t$$\\end{document} are distinct for all t, they form a braid B2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_2$$\\end{document} on n-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n-1$$\\end{document} strands. In this paper we study the relationship between B1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_1$$\\end{document} and B2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_2$$\\end{document}. Composing the polynomials gt\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g_t$$\\end{document} with the argument map defines a pseudo-fibration map on the complement of the closure of B1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_1$$\\end{document} in C×S1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {C}}\ imes S^1$$\\end{document}, whose critical points lie on B2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_2$$\\end{document}. We prove that for B1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_1$$\\end{document} a T-homogeneous braid and B2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_2$$\\end{document} the trivial braid this map can be taken to be a fibration map. In the case of homogeneous braids we present a visualization of this fact. Our work implies that for every pair of links L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_1$$\\end{document} and L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_2$$\\end{document} there is a mixed polynomial f:C2→C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f:{\\mathbb {C}}^2\\rightarrow {\\mathbb {C}}$$\\end{document} in complex variables u, v and the complex conjugate v¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\overline{v}$$\\end{document} such that both f and the derivative fu\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f_u$$\\end{document} have a weakly isolated singularity at the origin with L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_1$$\\end{document} as the link of the singularity of f and L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_2$$\\end{document} as a sublink of the link of the singularity of fu\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f_u$$\\end{document}.

CHARACTERISATION OF PRIMES DIVIDING THE INDEX OF A CLASS OF POLYNOMIALS AND ITS APPLICATIONS

Abstract Let ${\mathbb {Z}}_{K}$ denote the ring of algebraic integers of an algebraic number field $K = {\mathbb Q}(\theta )$ , where $\theta $ is a root of a monic irreducible polynomial $f(x) = x^n + a(bx+c)^m \in {\mathbb {Z}}[x]$ , $1\leq m&lt;n$ . We say $f(x)$ is monogenic if $\{1, \theta , \ldots , \theta ^{n-1}\}$ is a basis for ${\mathbb {Z}}_K$ . We give necessary and sufficient conditions involving only $a, b, c, m, n$ for $f(x)$ to be monogenic. Moreover, we characterise all the primes dividing the index of the subgroup ${\mathbb {Z}}[\theta ]$ in ${\mathbb {Z}}_K$ . As an application, we also provide a class of monogenic polynomials having non square-free discriminant and Galois group $S_n$ , the symmetric group on n letters.

Distribution of constant terms of irreducible polynomials in ℤ_p[x] whose degree is a product of two distinct odd primes

We obtain explicit formulas for the number of monic irreducible polynomials with prescribed constant term and degree $q_1q_2$ over a finite field, where $q_1$ and $q_2$ are distinct odd~primes. These formulas are derived from work done by Yucas. We show that the number of polynomials of a given constant term depends only on whether the constant term is a $q_1$-residue and/or a $q_2$-residue in the underlying field. We further show that as $k$ becomes large, the proportion of irreducible polynomials having each constant term is asymptotically equal. This paper continues work done in [1].