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1. CMB Online first

Rousseau, C.
 The Bifurcation Diagram of Cubic Polynomial Vector Fields on $\mathbb C\mathbb P^1$ In this paper we give the bifurcation diagram of the family of cubic vector fields $\dot z=z^3+ \epsilon_1z+\epsilon_0$ for $z\in \mathbb{C}\mathbb{P}^1$, depending on the values of $\epsilon_1,\epsilon_0\in\mathbb{C}$. The bifurcation diagram is in $\mathbb{R}^4$, but its conic structure allows describing it for parameter values in $\mathbb{S}^3$. There are two open simply connected regions of structurally stable vector fields separated by surfaces corresponding to bifurcations of homoclinic connections between two separatrices of the pole at infinity. These branch from the codimension 2 curve of double singular points. We also explain the bifurcation of homoclinic connection in terms of the description of Douady and Sentenac of polynomial vector fields. Keywords:complex polynomial vector field, bifurcation diagram, Douady-Sentenac invariantCategories:34M45, 32G34

2. CMB Online first

Friedl, Stefan; Vidussi, Stefano
 Twisted Alexander invariants detect trivial links It follows from earlier work of Silver--Williams and the authors that twisted Alexander polynomials detect the unknot and the Hopf link. We now show that twisted Alexander polynomials also detect the trefoil and the figure-8 knot, that twisted Alexander polynomials detect whether a link is split and that twisted Alexander modules detect trivial links. We use this result to provide algorithms for detecting whether a link is the unlink, whether it is split and whether it is totally split. Keywords:twisted Alexander polynomial, virtual fibering theorem, unlink detectionCategory:57M27

3. CMB Online first

Kurdyka, Krzysztof; Paunescu, Laurentiu
 Nuij type pencils of hyperbolic polynomials Nuij's theorem states that if a polynomial $p\in \mathbb{R}[z]$ is hyperbolic (i.e. has only real roots) then $p+sp'$ is also hyperbolic for any $s\in \mathbb{R}$. We study other perturbations of hyperbolic polynomials of the form $p_a(z,s): =p(z) +\sum_{k=1}^d a_ks^kp^{(k)}(z)$. We give a full characterization of those $a= (a_1, \dots, a_d) \in \mathbb{R}^d$ for which $p_a(z,s)$ is a pencil of hyperbolic polynomials. We give also a full characterization of those $a= (a_1, \dots, a_d) \in \mathbb{R}^d$ for which the associated families $p_a(z,s)$ admit universal determinantal representations. In fact we show that all these sequences come from special symmetric Toeplitz matrices. Keywords:hyperbolic polynomial, stable polynomial, determinantal representa- tion, symmetric Toeplitz matrixCategories:15A15, 30C10, 47A56

4. CMB Online first

Dobrowolski, Edward
 A note on Lawton's theorem We prove Lawton's conjecture about the upper bound on the measure of the set on the unit circle on which a complex polynomial with a bounded number of coefficients takes small values. Namely, we prove that Lawton's bound holds for polynomials that are not necessarily monic. We also provide an analogous bound for polynomials in several variables. Finally, we investigate the dependence of the bound on the multiplicity of zeros for polynomials in one variable. Keywords:polynomial, Mahler measureCategories:11R09, 11R06

5. CMB Online first

Eroǧlu, Münevver Pınar; Argaç, Nurcan
 On Identities with Composition of Generalized Derivations Let $R$ be a prime ring with extended centroid $C$, $Q$ maximal right ring of quotients of $R$, $RC$ central closure of $R$ such that $dim_{C}(RC) \gt 4$, $f(X_{1},\dots,X_{n})$ a multilinear polynomial over $C$ which is not central-valued on $R$ and $f(R)$ the set of all evaluations of the multilinear polynomial $f\big(X_{1},\dots,X_{n}\big)$ in $R$. Suppose that $G$ is a nonzero generalized derivation of $R$ such that $G^2\big(u\big)u \in C$ for all $u\in f(R)$ then one of the following conditions holds: (I) there exists $a\in Q$ such that $a^2=0$ and either $G(x)=ax$ for all $x\in R$ or $G(x)=xa$ for all $x\in R$; (II) there exists $a\in Q$ such that $0\neq a^2\in C$ and either $G(x)=ax$ for all $x\in R$ or $G(x)=xa$ for all $x\in R$ and $f(X_{1},\dots,X_{n})^{2}$ is central-valued on $R$; (III) $char(R)=2$ and one of the following holds: (i) there exist $a, b\in Q$ such that $G(x)=ax+xb$ for all $x\in R$ and $a^{2}=b^{2}\in C$; (ii) there exist $a, b\in Q$ such that $G(x)=ax+xb$ for all $x\in R$, $a^{2}, b^{2}\in C$ and $f(X_{1},\ldots,X_{n})^{2}$ is central-valued on $R$; (iii) there exist $a \in Q$ and an $X$-outer derivation $d$ of $R$ such that $G(x)=ax+d(x)$ for all $x\in R$, $d^2=0$ and $a^2+d(a)=0$; (iv) there exist $a \in Q$ and an $X$-outer derivation $d$ of $R$ such that $G(x)=ax+d(x)$ for all $x\in R$, $d^2=0$, $a^2+d(a)\in C$ and $f(X_{1},\dots,X_{n})^{2}$ is central-valued on $R$. Moreover, we characterize the form of nonzero generalized derivations $G$ of $R$ satisfying $G^2(x)=\lambda x$ for all $x\in R$, where $\lambda \in C$. Keywords:prime ring, generalized derivation, composition, extended centroid, multilinear polynomial, maximal right ring of quotientsCategories:16N60, 16N25

6. CMB 2016 (vol 60 pp. 77)

Christ, Michael; Rieffel, Marc A.
 Nilpotent Group C*-algebras as Compact Quantum Metric Spaces Let $\mathbb{L}$ be a length function on a group $G$, and let $M_\mathbb{L}$ denote the operator of pointwise multiplication by $\mathbb{L}$ on $\lt(G)$. Following Connes, $M_\mathbb{L}$ can be used as a Dirac'' operator for the reduced group C*-algebra $C_r^*(G)$. It defines a Lipschitz seminorm on $C_r^*(G)$, which defines a metric on the state space of $C_r^*(G)$. We show that for any length function satisfying a strong form of polynomial growth on a discrete group, the topology from this metric coincides with the weak-$*$ topology (a key property for the definition of a compact quantum metric space''). In particular, this holds for all word-length functions on finitely generated nilpotent-by-finite groups. Keywords:group C*-algebra, Dirac operator, quantum metric space, discrete nilpotent group, polynomial growthCategories:46L87, 20F65, 22D15, 53C23, 58B34

7. CMB 2016 (vol 59 pp. 794)

Hashemi, Ebrahim; Amirjan, R.
 Zero-divisor Graphs of Ore Extensions over Reversible Rings Let $R$ be an associative ring with identity. First we prove some results about zero-divisor graphs of reversible rings. Then we study the zero-divisors of the skew power series ring $R[[x;\alpha]]$, whenever $R$ is reversible and $\alpha$-compatible. Moreover, we compare the diameter and girth of the zero-divisor graphs of $\Gamma(R)$, $\Gamma(R[x;\alpha,\delta])$ and $\Gamma(R[[x;\alpha]])$, when $R$ is reversible and $(\alpha,\delta)$-compatible. Keywords:zero-divisor graphs, reversible rings, McCoy rings, polynomial rings, power series ringsCategories:13B25, 05C12, 16S36

8. CMB 2016 (vol 59 pp. 661)

Ying, Zhiling; Koşan, Tamer; Zhou, Yiqiang
 Rings in Which Every Element is a Sum of Two Tripotents Let $R$ be a ring. The following results are proved: $(1)$ every element of $R$ is a sum of an idempotent and a tripotent that commute iff $R$ has the identity $x^6=x^4$ iff $R\cong R_1\times R_2$, where $R_1/J(R_1)$ is Boolean with $U(R_1)$ a group of exponent $2$ and $R_2$ is zero or a subdirect product of $\mathbb Z_3$'s; $(2)$ every element of $R$ is either a sum or a difference of two commuting idempotents iff $R\cong R_1\times R_2$, where $R_1/J(R_1)$ is Boolean with $J(R_1)=0$ or $J(R_1)=\{0,2\}$, and $R_2$ is zero or a subdirect product of $\mathbb Z_3$'s; $(3)$ every element of $R$ is a sum of two commuting tripotents iff $R\cong R_1\times R_2\times R_3$, where $R_1/J(R_1)$ is Boolean with $U(R_1)$ a group of exponent $2$, $R_2$ is zero or a subdirect product of $\mathbb Z_3$'s, and $R_3$ is zero or a subdirect product of $\mathbb Z_5$'s. Keywords:idempotent, tripotent, Boolean ring, polynomial identity $x^3=x$, polynomial identity $x^6=x^4$, polynomial identity $x^8=x^4$Categories:16S50, 16U60, 16U90

9. CMB 2016 (vol 59 pp. 340)

Kȩpczyk, Marek
 A Note on Algebras that are Sums of Two Subalgebras We study an associative algebra $A$ over an arbitrary field, that is a sum of two subalgebras $B$ and $C$ (i.e. $A=B+C$). We show that if $B$ is a right or left Artinian $PI$ algebra and $C$ is a $PI$ algebra, then $A$ is a $PI$ algebra. Additionally we generalize this result for semiprime algebras $A$. Consider the class of all semisimple finite dimensional algebras $A=B+C$ for some subalgebras $B$ and $C$ which satisfy given polynomial identities $f=0$ and $g=0$, respectively. We prove that all algebras in this class satisfy a common polynomial identity. Keywords:rings with polynomial identities, prime ringsCategories:16N40, 16R10, , 16S36, 16W60, 16R20

10. CMB 2015 (vol 58 pp. 818)

Llibre, Jaume; Zhang, Xiang
 On the Limit Cycles of Linear Differential Systems with Homogeneous Nonlinearities We consider the class of polynomial differential systems of the form $\dot x= \lambda x-y+P_n(x,y)$, $\dot y=x+\lambda y+ Q_n(x,y),$ where $P_n$ and $Q_n$ are homogeneous polynomials of degree $n$. For this class of differential systems we summarize the known results for the existence of limit cycles, and we provide new results for their nonexistence and existence. Keywords:polynomial differential system, limit cycles, differential equations on the cylinderCategories:34C35, 34D30

11. CMB 2015 (vol 58 pp. 704)

Benamar, H.; Chandoul, A.; Mkaouar, M.
 On the Continued Fraction Expansion of Fixed Period in Finite Fields The Chowla conjecture states that, if $t$ is any given positive integer, there are infinitely many prime positive integers $N$ such that $\operatorname{Per} (\sqrt{N})=t$, where $\operatorname{Per} (\sqrt{N})$ is the period length of the continued fraction expansion for $\sqrt{N}$. C. Friesen proved that, for any $k\in \mathbb{N}$, there are infinitely many square-free integers $N$, where the continued fraction expansion of $\sqrt{N}$ has a fixed period. In this paper, we describe all polynomials $Q\in \mathbb{F}_q[X]$ for which the continued fraction expansion of $\sqrt {Q}$ has a fixed period, also we give a lower bound of the number of monic, non-squares polynomials $Q$ such that $\deg Q= 2d$ and $Per \sqrt {Q}=t$. Keywords:continued fractions, polynomials, formal power seriesCategories:11A55, 13J05

12. CMB 2015 (vol 58 pp. 877)

Zaatra, Mohamed
 Generating Some Symmetric Semi-classical Orthogonal Polynomials We show that if $v$ is a regular semi-classical form (linear functional), then the symmetric form $u$ defined by the relation $x^{2}\sigma u = -\lambda v$, where $(\sigma f)(x)=f(x^{2})$ and the odd moments of $u$ are $0$, is also regular and semi-classical form for every complex $\lambda$ except for a discrete set of numbers depending on $v$. We give explicitly the three-term recurrence relation and the structure relation coefficients of the orthogonal polynomials sequence associated with $u$ and the class of the form $u$ knowing that of $v$. We conclude with an illustrative example. Keywords:orthogonal polynomials, quadratic decomposition, semi-classical forms, structure relationCategories:33C45, 42C05

13. CMB 2015 (vol 59 pp. 159)

MacColl, Joseph
 Rotors in Khovanov Homology Anstee, Przytycki, and Rolfsen introduced the idea of rotants, pairs of links related by a generalised form of link mutation. We exhibit infinitely many pairs of rotants which can be distinguished by Khovanov homology, but not by the Jones polynomial. Keywords:geometric topology, knot theory, rotants, khovanov homology, jones polynomialCategories:57M27, 57M25

14. CMB 2015 (vol 58 pp. 423)

Yamagishi, Masakazu
 Resultants of Chebyshev Polynomials: The First, Second, Third, and Fourth Kinds We give an explicit formula for the resultant of Chebyshev polynomials of the first, second, third, and fourth kinds. We also compute the resultant of modified cyclotomic polynomials. Keywords:resultant, Chebyshev polynomial, cyclotomic polynomialCategories:11R09, 11R18, 12E10, 33C45

15. CMB 2015 (vol 58 pp. 225)

Aghigh, Kamal; Nikseresht, Azadeh
 Characterizing Distinguished Pairs by Using Liftings of Irreducible Polynomials Let $v$ be a henselian valuation of any rank of a field $K$ and $\overline{v}$ be the unique extension of $v$ to a fixed algebraic closure $\overline{K}$ of $K$. In 2005, it was studied properties of those pairs $(\theta,\alpha)$ of elements of $\overline{K}$ with $[K(\theta): K]\gt [K(\alpha): K]$ where $\alpha$ is an element of smallest degree over $K$ such that $$\overline{v}(\theta-\alpha)=\sup\{\overline{v}(\theta-\beta) |\ \beta\in \overline{K}, \ [K(\beta): K]\lt [K(\theta): K]\}.$$ Such pairs are referred to as distinguished pairs. We use the concept of liftings of irreducible polynomials to give a different characterization of distinguished pairs. Keywords:valued fields, non-Archimedean valued fields, irreducible polynomialsCategories:12J10, 12J25, 12E05

16. CMB 2014 (vol 57 pp. 609)

Nasr-Isfahani, Alireza
 Jacobson Radicals of Skew Polynomial Rings of Derivation Type We provide necessary and sufficient conditions for a skew polynomial ring of derivation type to be semiprimitive, when the base ring has no nonzero nil ideals. This extends existing results on the Jacobson radical of skew polynomial rings of derivation type. Keywords:skew polynomial rings, Jacobson radical, derivationCategories:16S36, 16N20

17. CMB 2014 (vol 57 pp. 538)

Ide, Joshua; Jones, Lenny
 Infinite Families of $A_4$-Sextic Polynomials In this article we develop a test to determine whether a sextic polynomial that is irreducible over $\mathbb{Q}$ has Galois group isomorphic to the alternating group $A_4$. This test does not involve the computation of resolvents, and we use this test to construct several infinite families of such polynomials. Keywords:Galois group, sextic polynomial, inverse Galois theory, irreducible polynomialCategories:12F10, 12F12, 11R32, 11R09

18. CMB 2012 (vol 56 pp. 759)

Issa, Zahraa; Lalín, Matilde
 A Generalization of a Theorem of Boyd and Lawton The Mahler measure of a nonzero $n$-variable polynomial $P$ is the integral of $\log|P|$ on the unit $n$-torus. A result of Boyd and Lawton says that the Mahler measure of a multivariate polynomial is the limit of Mahler measures of univariate polynomials. We prove the analogous result for different extensions of Mahler measure such as generalized Mahler measure (integrating the maximum of $\log|P|$ for possibly different $P$'s), multiple Mahler measure (involving products of $\log|P|$ for possibly different $P$'s), and higher Mahler measure (involving $\log^k|P|$). Keywords:Mahler measure, polynomialCategories:11R06, 11R09

19. CMB 2012 (vol 56 pp. 844)

Shparlinski, Igor E.
 On the Average Number of Square-Free Values of Polynomials We obtain an asymptotic formula for the number of square-free integers in $N$ consecutive values of polynomials on average over integral polynomials of degree at most $k$ and of height at most $H$, where $H \ge N^{k-1+\varepsilon}$ for some fixed $\varepsilon\gt 0$. Individual results of this kind for polynomials of degree $k \gt 3$, due to A. Granville (1998), are only known under the $ABC$-conjecture. Keywords:polynomials, square-free numbersCategory:11N32

20. CMB 2012 (vol 56 pp. 602)

Louboutin, Stéphane R.
 Resultants of Chebyshev Polynomials: A Short Proof We give a simple proof of the value of the resultant of two Chebyshev polynomials (of the first or the second kind), values lately obtained by D. P. Jacobs, M. O. Rayes and V. Trevisan. Keywords:resultant, Chebyshev polynomials, cyclotomic polynomialsCategories:11R09, 11R04

21. CMB 2012 (vol 56 pp. 769)

Lahiri, Indrajit; Kaish, Imrul
 A Non-zero Value Shared by an Entire Function and its Linear Differential Polynomials In this paper we study uniqueness of entire functions sharing a non-zero finite value with linear differential polynomials and address a result of W. Wang and P. Li. Keywords:entire function, linear differential polynomial, value sharingCategory:30D35

22. CMB 2012 (vol 56 pp. 584)

Liau, Pao-Kuei; Liu, Cheng-Kai
 On Automorphisms and Commutativity in Semiprime Rings Let $R$ be a semiprime ring with center $Z(R)$. For $x,y\in R$, we denote by $[x,y]=xy-yx$ the commutator of $x$ and $y$. If $\sigma$ is a non-identity automorphism of $R$ such that $$\Big[\big[\dots\big[[\sigma(x^{n_0}),x^{n_1}],x^{n_2}\big],\dots\big],x^{n_k}\Big]=0$$ for all $x \in R$, where $n_{0},n_{1},n_{2},\dots,n_{k}$ are fixed positive integers, then there exists a map $\mu\colon R\rightarrow Z(R)$ such that $\sigma(x)=x+\mu(x)$ for all $x\in R$. In particular, when $R$ is a prime ring, $R$ is commutative. Keywords:automorphism, generalized polynomial identity (GPI)Categories:16N60, 16W20, 16R50

23. CMB 2011 (vol 56 pp. 510)

Dubickas, Artūras
 Linear Forms in Monic Integer Polynomials We prove a necessary and sufficient condition on the list of nonzero integers $u_1,\dots,u_k$, $k \geq 2$, under which a monic polynomial $f \in \mathbb{Z}[x]$ is expressible by a linear form $u_1f_1+\dots+u_kf_k$ in monic polynomials $f_1,\dots,f_k \in \mathbb{Z}[x]$. This condition is independent of $f$. We also show that if this condition holds, then the monic polynomials $f_1,\dots,f_k$ can be chosen to be irreducible in $\mathbb{Z}[x]$. Keywords:irreducible polynomial, height, linear form in polynomials, Eisenstein's criterionCategories:11R09, 11C08, 11B83

24. CMB 2011 (vol 56 pp. 251)

Borwein, Peter; Choi, Stephen K. K.; Ganguli, Himadri
 Sign Changes of the Liouville Function on Quadratics Let $\lambda (n)$ denote the Liouville function. Complementary to the prime number theorem, Chowla conjectured that \begin{equation*} \label{a.1} \sum_{n\le x} \lambda (f(n)) =o(x)\tag{$*$} \end{equation*} for any polynomial $f(x)$ with integer coefficients which is not of form $bg(x)^2$. When $f(x)=x$, $(*)$ is equivalent to the prime number theorem. Chowla's conjecture has been proved for linear functions, but for degree greater than 1, the conjecture seems to be extremely hard and remains wide open. One can consider a weaker form of Chowla's conjecture. Conjecture 1. [Cassaigne et al.] If $f(x) \in \mathbb{Z} [x]$ and is not in the form of $bg^2(x)$ for some $g(x)\in \mathbb{Z}[x]$, then $\lambda (f(n))$ changes sign infinitely often. Clearly, Chowla's conjecture implies Conjecture 1. Although weaker, Conjecture 1 is still wide open for polynomials of degree $\gt 1$. In this article, we study Conjecture 1 for quadratic polynomials. One of our main theorems is the following. Theorem 1 Let $f(x) = ax^2+bx +c$ with $a\gt 0$ and $l$ be a positive integer such that $al$ is not a perfect square. If the equation $f(n)=lm^2$ has one solution $(n_0,m_0) \in \mathbb{Z}^2$, then it has infinitely many positive solutions $(n,m) \in \mathbb{N}^2$. As a direct consequence of Theorem 1, we prove the following. Theorem 2 Let $f(x)=ax^2+bx+c$ with $a \in \mathbb{N}$ and $b,c \in \mathbb{Z}$. Let $A_0=\Bigl[\frac{|b|+(|D|+1)/2}{2a}\Bigr]+1.$ Then either the binary sequence $\{ \lambda (f(n)) \}_{n=A_0}^\infty$ is a constant sequence or it changes sign infinitely often. Some partial results of Conjecture 1 for quadratic polynomials are also proved using Theorem 1. Keywords:Liouville function, Chowla's conjecture, prime number theorem, binary sequences, changes sign infinitely often, quadratic polynomials, Pell equationCategories:11N60, 11B83, 11D09

25. CMB 2011 (vol 56 pp. 194)

Stefánsson, Úlfar F.
 On the Smallest and Largest Zeros of MÃ¼ntz-Legendre Polynomials MÃ¼ntz-Legendre polynomials $L_n(\Lambda;x)$ associated with a sequence $\Lambda=\{\lambda_k\}$ are obtained by orthogonalizing the system $(x^{\lambda_0}, x^{\lambda_1}, x^{\lambda_2}, \dots)$ in $L_2[0,1]$ with respect to the Legendre weight. If the $\lambda_k$'s are distinct, it is well known that $L_n(\Lambda;x)$ has exactly $n$ zeros $l_{n,n}\lt l_{n-1,n}\lt \cdots \lt l_{2,n}\lt l_{1,n}$ on $(0,1)$. First we prove the following global bound for the smallest zero, $$\exp\biggl(-4\sum_{j=0}^n \frac{1}{2\lambda_j+1}\biggr) \lt l_{n,n}.$$ An important consequence is that if the associated MÃ¼ntz space is non-dense in $L_2[0,1]$, then $$\inf_{n}x_{n,n}\geq \exp\biggl({-4\sum_{j=0}^{\infty} \frac{1}{2\lambda_j+1}}\biggr)\gt 0,$$ so the elements $L_n(\Lambda;x)$ have no zeros close to 0. Furthermore, we determine the asymptotic behavior of the largest zeros; for $k$ fixed, $$\lim_{n\rightarrow\infty} \vert \log l_{k,n}\vert \sum_{j=0}^n (2\lambda_j+1)= \Bigl(\frac{j_k}{2}\Bigr)^2,$$ where $j_k$ denotes the $k$-th zero of the Bessel function $J_0$. Keywords:MÃ¼ntz polynomials, MÃ¼ntz-Legendre polynomialsCategories:42C05, 42C99, 41A60, 30B50
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