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The concept of unity roots plays a central role in the theory of field extensions and polynomials roots' computing. The objective of this paper is to find the algebraic formula for computing the symbolic 2-plithogenic and 3-plithogenic complex roots of unity, where a general formula will be provided with many related examples up to the exponent 3. [ABSTRACT FROM AUTHOR]

We study the complexity of approximating the value of the independent set polynomial ZG(\lambda) of a graph G with maximum degree \Delta when the activity \lambda is a complex number. When \lambda is real, the complexity picture is well understood, and is captured by two real-valued thresholds \lambda \ast and \lambda c, which depend on \Delta and satisfy 0 < \lambda \ast < \lambda c. It is known that if \lambda is a real number in the interval (\lambda \ast, \lambda c) then there is a fully polynomial time approximation scheme (FPTAS) for approximating ZG(\lambda) on graphs G with maximum degree at most \Delta. On the other hand, if \lambda is a real number outside of the (closed) interval, then approximation is NP-hard. The key to establishing this picture was the interpretation of the thresholds \lambda \ast and \lambda c on the \Delta -regular tree. The ''occupation ratio"" of a \Delta -regular tree T is the contribution to ZT (\lambda) from independent sets containing the root of the tree, divided by ZT (\lambda) itself. This occupation ratio converges to a limit, as the height of the tree grows, if and only if \lambda \in [ \lambda \ast, \lambda c]. Unsurprisingly, the case where \lambda is complex is more challenging. It is known that there is an FPTAS when \lambda is a complex number with norm at most \lambda \ast and also when \lambda is in a small strip surrounding the real interval [0, \lambda c). However, neither of these results is believed to fully capture the truth about when approximation is possible. Peters and Regts identified the complex values of \lambda for which the occupation ratio of the \Delta -regular tree converges. These values carve a cardioid-shaped region \Lambda \Delta in the complex plane, whose boundary includes the critical points \lambda \ast and \lambda c. Motivated by the picture in the real case, they asked whether \Lambda \Delta marks the true approximability threshold for general complex values \lambda. Our main result shows that for every \lambda outside of \Lambda \Delta, the problem of approximating ZG(\lambda) on graphs G with maximum degree at most \Delta is indeed NP-hard. In fact, when \lambda is outside of \Lambda \Delta and is not a positive real number, we give the stronger result that approximating ZG(\lambda) is actually \#P-hard. Further, on the negative real axis, when \lambda < \lambda \ast, we show that it is \#P-hard to even decide whether ZG(\lambda) > 0, resolving in the affirmative a conjecture of Harvey, Srivastava, and Vondrak. Our proof techniques are based around tools from complex analysis--specifically the study of iterative multivariate rational maps. [ABSTRACT FROM AUTHOR]

The polar derivative of a polynomial P(z) of degree n with respect to a complex number γ is a polynomial nP(z) + (γ-z)P′(z) of degree at most n-1 and is denoted by DγP(z). We consider the class of polynomials P(z) = a0 + Σv=μN avzv, μ ≥ 1, of degree n such that P(z) ≠0 in |z| < k, k ≥ 1 and establish some upper bound estimates for the maximum modulus of DγP(z) on the unit disk by involving some of the coefficients of P(z). The obtained results refine and generalize some well known polynomial inequalities. [ABSTRACT FROM AUTHOR]

A complex unit gain graph is a simple graph in which each orientation of an edge is given a complex number with modulus 1 and its inverse is assigned to the opposite orientation of the edge. In this article, first we establish bounds for the eigenvalues of the complex unit gain graphs. Then we study some of the properties of the adjacency matrix of a complex unit gain graph in connection with the characteristic and the permanental polynomials. Then we establish spectral properties of the adjacency matrices of complex unit gain graphs. In particular, using Perron–Frobenius theory, we establish a characterization for bipartite graphs in terms of the set of eigenvalues of a gain graph and the set of eigenvalues of the underlying graph. Also, we derive an equivalent condition on the gain so that the eigenvalues of the gain graph and the eigenvalues of the underlying graph are the same. [ABSTRACT FROM AUTHOR]

All the roots of the general nth degree trinomial admit certain convenient representations in terms of the Lambert and Euler series for the asymmetric and symmetric cases of the trinomial equation, respectively. Previously, various methods have been used to provide the proofs for the general terms of these two series. Taking n to be any real or complex number, we presently give an alternative proof using the Bell (or exponential) polynomials. The ensuing series is summed up yielding a single, compact, explicit, analytical formula for all the trinomial roots as the confluent Fox-Wright function 1Ψ1. Moreover, we also derive a slightly different, single formula of the trinomial root raised to any power (real or complex number) as another 1Ψ1 function. Further, in this study, the logarithm of the trinomial root is likewise expressed through a single, concise series with the binomial expansion coefficients or the Pochhammer symbols. These findings are anticipated to be of considerable help in various applications of trinomial roots. Namely, several properties of the 1Ψ1 function can advantageously be employed for its implementations in practice. For example, the simple expressions for the asymptotic limits of the 1Ψ1 function at both small and large values of the independent variable can be used to readily predict, by analytical means, the critical behaviors of the studied system in the two extreme conditions. Such limiting situations can be e.g. at the beginning of the time evolution of a system, and in the distant future, if the independent variable is time, or at low and high doses when the independent variable is radiation dose, etc. The present analytical solutions for the trinomial roots are numerically illustrated in the genome multiplicity corrections for survival of synchronous cell populations after irradiation. [ABSTRACT FROM AUTHOR]

In many stability problems, the investigation of pure imaginary roots for a polynomial family is very important. Given a pure imaginary complex number, the set of all images of uncertainty vectors is called the value set corresponding to this pure imaginary complex number. The question whether these sets contain the origin is very important from robust stability point of view of a polynomial family. Cutoff frequency guarantees the noninclusion of the origin to the value set for large frequencies. In this paper, we give a procedure for more strict estimation of cutoff frequency and applications of the obtained result to the constant inertia problem of a polynomial family. [ABSTRACT FROM AUTHOR]

Let n ≥ 2 be a fixed integer, and denote by M n the algebra of all n × n complex matrices. Fix also an integer k such that 1 ≤ k < n. We prove that if φ : M n → M n is a linear map which preserves the polynomial numerical radius of order k , there exist then a unitary n × n complex matrix U and a complex number ξ of modulus one such that either φ (T) = ξ U ⁎ T U for all T ∈ M n , or φ (T) = ξ U ⁎ T t U for all T ∈ M n. [ABSTRACT FROM AUTHOR]

The purpose of this article is to generalize the ring of q-Appell polynomials to the complex case. The formulas for q-Appell polynomials thus appear again, with similar names, in a purely symmetric way. Since these complex q-Appell polynomials are also q-complex analytic functions, we are able to give a first example of the q-Cauchy-Riemann equations. Similarly, in the spirit of Kim and Ryoo, we can define q-complex Bernoulli and Euler polynomials. Previously, in order to obtain the q-Appell polynomial, we would make a q-addition of the corresponding q-Appell number with x. This is now replaced by a q-addition of the corresponding q-Appell number with two infinite function sequences C;q(x; y) and S;q(x; y) for the real and imaginary part of a new so-called q-complex number appearing in the generating function. Finally, we can prove q-analogues of the Cauchy-Riemann equations. [ABSTRACT FROM AUTHOR]

Let P(z) be a polynomial of degree nn and for any complex number aa, let DαP(z)=nP(z)+(α-z)P′(z) denote the polar derivative of P(z) with respect to a complex number α. In this paper, we present an integral inequality for the polar derivative of a polynomial P(z). Our result includes as special cases several interesting generalizations of some Zygmund type inequalities for polynomials. [ABSTRACT FROM AUTHOR]

Abstract We consider the problem of scheduling complex-valued demands over a discretized time horizon. Given a set of users, each user is associated with a set of demands representing different power consumption preferences. A demand is represented by a complex number, a time interval, and a utility value obtained if it is satisfied. At each time slot, the magnitude of the total selected demands should not exceed a given generation capacity. This naturally captures the supply constraints in alternating current (AC) electric systems. In this paper, we consider maximizing the aggregate user utility subject to power supply limits over a time horizon. We present approximation algorithms characterized by the maximum angle ϕ between any two complex-valued demands. More precisely, a PTAS is presented for the case ϕ ∈ [ 0 , π 2 ] , a bi-criteria FPTAS for ϕ ∈ [ 0 , π - ε ] for any polynomially small ε , assuming the number of time slots in the discretized time horizon is a constant. Furthermore, if the number of time slots is part of the input, we present a reduction to the real-valued unsplittable flow problem on a path with only a constant approximation ratio. Finally, we present a practical greedy algorithm for the single time slot case with an approximation ratio of 1 2 cos ⁡ ϕ 2 and a running time complexity of only O (N log ⁡ N) , N standing for the aggregate number of user demands, which can be implemented efficiently in practice. [ABSTRACT FROM AUTHOR]

Abstract It is well known that for two univariate polynomials over the complex number field the number of their common roots is equal to the order of their resultant. In this paper, we show that this fundamental relationship still holds for the tropical polynomials under suitable adaptation of the notion of order, if the roots are simple and non-zero. [ABSTRACT FROM AUTHOR]

We consider König’s method for finding roots of polynomials. Let K f , n be the function defined in König’s method for polynomial f of order n ( n ≥ 2 ) . For any given complex number λ ∈ C and any given set Ω ≔ { z 1 , z 2 , … , z k } of k ( k ≥ 2 ) distinct complex numbers, we present a procedure of constructing a polynomial f n , λ for which Ω is a k -cycle of K f n , λ , n with multiplier λ . [ABSTRACT FROM AUTHOR]

Let P ( z ) be a polynomial of degree n and for any complex number α , let D α P ( z ) : = n P ( z ) + ( α − z ) P ′ ( z ) denote the polar derivative of P ( z ) with respect to α . In this paper, we present an integral inequality for the polar derivative of a polynomial. Our theorem includes as special cases several interesting generalisations and refinements of Erdöx–Lax theorem. [ABSTRACT FROM AUTHOR]

We prove that if P( z) is a polynomial of degree n with zeros z , that satisfy $${|z_m| \geq K_m \geq 1}$$ , $${1 \leq m \leq n}$$ , then for any p > 0, and for every complex number α, with $${|\alpha| \geq 1}$$ , we have where $${G_{p}=\big\{\frac{2\pi}{\int_0^{2\pi}|t_0+e^{i\theta}|^{p}\,d\theta}\big\}^{{1}/{p}}}$$ , and $${t_0=\big\{1+\frac{n}{\sum_{m=1}^n\frac{1}{K_m-1}}\big\}}$$ if $${K_{m}>1}$$ $${(1\leq m \leq n)}$$ , and t = 1 if K = 1 for some m, $${1\leq m\leq n}$$ . Our results generalize and sharpen several of the known results. [ABSTRACT FROM AUTHOR]

Recently, M. Abért and T. Hubai studied the following problem. The chromatic measure of a finite simple graph is defined to be the uniform distribution on its chromatic roots. Abért and Hubai proved that for a Benjamini–Schramm convergent sequence of finite graphs, the chromatic measures converge in holomorphic moments. They also showed that the normalized logarithm of the chromatic polynomial converges to a harmonic real function outside a bounded disc. In this paper we generalize their work to a wide class of graph polynomials, namely, multiplicative graph polynomials of bounded exponential type. A special case of our results is that for any fixed complex number v 0 the measures arising from the Tutte polynomial Z G n ( z , v 0 ) converge in holomorphic moments if the sequence ( G n ) of finite graphs is Benjamini–Schramm convergent. This answers a question of Abért and Hubai in the affirmative. Even in the original case of the chromatic polynomial, our proof is considerably simpler. [ABSTRACT FROM AUTHOR]

A Gaussian integer is a complex number whose real and imaginary parts are both integers. Meanwhile, a sequence is defined as perfect if and only if it has an ideal periodic auto-correlation function. This paper proposes a method for constructing sparse perfect Gaussian integer sequences (SPGISs) in which most of the sequence elements are zero. The proposed SPGISs are obtained by linearly combining four base sequences or their cyclic-shift equivalents using nonzero Gaussian integer coefficients of equal magnitudes. Each base sequence contains four nonzero elements belonging to the set \\pm 1, \pm j\. The number of nonzero elements of the constructed SPGISs depends on the choice of complex coefficients and cyclic shifts. However, each SPGIS has at most 16 nonzero elements, irrespective of the sequence length. A systematic investigation is performed into the properties of the SPGISs and their Fourier dual equivalents. Finally, a general expression is derived for a perfect Gaussian integer sequence (PGIS) of length $4n$, where $n$ is any positive integer and most of the sequence elements are nonzero. [ABSTRACT FROM AUTHOR]

A polynomial is called unimodular if each of its coefficients is a complex number of modulus 1. A polynomial P of the form P ( z ) = ∑ j = 0 n a j z j is called conjugate reciprocal if a n − j = a ‾ j , a j ∈ C for each j = 0 , 1 , … , n . Let ∂ D be the unit circle of the complex plane. We prove that there is an absolute constant ε > 0 such that max z ∈ ∂ D ⁡ | f ( z ) | ≥ ( 1 + ε ) 4 / 3 m 1 / 2 , for every conjugate reciprocal unimodular polynomial of degree m . We also prove that there is an absolute constant ε > 0 such that M q ( f ′ ) ≤ exp ⁡ ( ε ( q − 2 ) / q ) 1 / 3 m 3 / 2 , 1 ≤ q < 2 , and M q ( f ′ ) ≥ exp ⁡ ( ε ( q − 2 ) / q ) 1 / 3 m 3 / 2 , 2 < q , for every conjugate reciprocal unimodular polynomial of degree m , where M q ( g ) = ( 1 2 π ∫ 0 2 π | g ( e i t ) | q d t ) 1 / q , q > 0 . [ABSTRACT FROM AUTHOR]

The article discusses the fundamental theorem of algebra. It mentions that every non-constant single-variable polynomial with complex coefficients has at least one complex root. It includes polynomials with real coefficients, since every real number is a complex number with an imaginary part equal to zero. It adds that every non-zero, single-variable, degree n polynomial with complex coefficients has, counted with multiplicity, exactly n roots.

Suppose we have a complex polynomial f ( z ) whose coefficients are inaccurate, and a prescribed complex number α such that f ( α ) ≠ 0 . We study the problem of computing a complex polynomial f ˜ ( z ) such that f ˜ ( α ) = 0 and the distance between f ˜ and f , i.e. ‖ f ˜ − f ‖ , is minimal. Considering that previous works usually took the usual l p -norm, weighted l p -norm and block-wise norm as distance measures, we first introduce a new-defined synthetic norm that integrates all these norms. Then, we propose a unified approach to study the proposed problem and succeed in giving explicit expressions of the nearest polynomial. The effectiveness of our approach is illustrated by two examples, one of which shows an extension of finding the nearest complex polynomial with a zero in a given domain. Finally, as an application of the new-defined norm, we discuss a matrix-valued optimization problem that is very common in machine learning. [ABSTRACT FROM AUTHOR]

Let $\mathscr{F}$ be a family of solutions of Laplace equations in a domain D and for each $f\in\mathscr{F}$, f has only zeros of multiplicity at least k. Let n be a positive integer and such that $n\geq\frac{1+\sqrt{1+4k(k+1)^{2}}}{2k}$. Let a be a complex number such that $a\neq0$. If for each pair of functions f and g in $\mathscr{F}$, $f^{n}f^{(k)}$ and $g^{n}g^{(k)}$ share a value in D, then $\mathscr{F}$ is normal in D. [ABSTRACT FROM AUTHOR]

This paper considers algebraic independence and hypertranscendence of functions satisfying Mahler-type functional equations $af(z^{r})=f(z)+R(z)$, where $a$ is a nonzero complex number, $r$ an integer greater than 1, and $R(z)$ a rational function. Well-known results from the scope of Mahler’s method then imply algebraic independence over the rationals of the values of these functions at algebraic points. As an application, algebraic independence results on reciprocal sums of Fibonacci and Lucas numbers are obtained. [ABSTRACT FROM AUTHOR]

Let P(z) be a polynomial of degree n having no zeros in |z| < k where k ≥ 1. Then it is known that for every real or complex number α with |α| ≥ 1, max |z|=1 |DαP(z)| ≤ n (|α| + k/1 + k) max |z|=1 |P(z)|, where DαP(z) = nP(z) + (α - z)P' (z) denotes the polar derivative of the polynomial P(z) of degree n with respect to a point α ∈ ℂ. In this paper, by a simple method, a refinement of the above inequality and other related results are obtained. [ABSTRACT FROM AUTHOR]

Using the basic fact that any formal power series over the real or complex number field can always be expressed in terms of given polynomials {p n (t)}, where p n (t) is of degree n, we extend the ordinary Riordan array (resp. Riordan group) to a generalized Riordan array (resp. generalized Riordan group) associated with {p n (t)}. As new application of the latter, a rather general Vandermonde-type convolution formula and certain of its particular forms are presented. The construction of the Abel type identities using the generalized Riordan arrays is also discussed. [ABSTRACT FROM AUTHOR]

Let f be a nonconstant entire function, let a be a finite nonzero complex number, and let L(f) = f(k) + ak-1f(k-1) + ... + a1f' + a0 f, where k ≥ 2 is a positive integer, aj (j = 0, 1, 2, ..., k-1) are constants. Suppose that f(z) = a ⇒ f'(z) = a, and f'(z) = a ⇒ L(f)(z) = L'(f)(z) = a. Then, if a0 ≠ 1, then either f(z) = Ae 1/1-a0 z + a or f(z) = Aez, where A is a nonzero constant; if a0 = 1, then L(f) = f. [ABSTRACT FROM AUTHOR]