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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]

Let P(z) be a polynomial of degree n and for any complex number α, 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 prove some Lr inequalities for the polar derivative of a polynomial have all zeros in |z| ≤ 1. Our theorem generalizes a result of Dewan and Mir [K. K. Dewan, A. Mir, Inequalities for the polar derivative of a polynomial, J. Interd. Math. 10 (2007), no. 4, 525–531] and includes as special cases several interesting many known results [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]

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]

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]

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]

Let p(z) be a polynomial of degree n. The polar derivative of p(z) with respect to a complex number α is defined by D α p (z) = n p (z) + (α - z) p ′ (z). If p (z) = z s ∑ j = 0 n - s c j z j , 0 ≤ s ≤ n , has all its zeros in | z | ≤ k , k ≥ 1 , then for | α | ≥ k , Kumar and Dhankhar [Bull, Math. Soc. Sci. Math., 63(4), 359-367 (2020)] proved max | z | = 1 | D α p (z) | ≥ n (| α | - k) 1 + k n - s 1 + (| c n - s | k n - | c 0 | k s) (k - 1) 2 (| c n - s | k n + | c 0 | k s + 1) max | z | = 1 | p (z) |. In this paper, we first improve the above inequality. Besides, we are able to prove an improvement of a result due to Govil and Mctume [Acta. Math. Hungar., 104, 115-126 (2004)] and also prove an inequality for a subclass of polynomials having no zero in | z | < k , k ≤ 1 . [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.

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]

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 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]

We introduce a new homomorphic encryption scheme that is natively capable of computing with complex numbers. This is done by generalizing recent work of Chen, Laine, Player and Xia, who modified the Fan–Vercauteren scheme by replacing the integral plaintext modulus t by a linear polynomial X − b. Our generalization studies plaintext moduli of the form Xm + b. Our construction significantly reduces the noise growth in comparison to the original FV scheme, so much deeper arithmetic circuits can be homomorphically executed. [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]

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]

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]

For a sequence (a n) (a n) of complex numbers we consider the cubic parabolic polynomials f n (z) = z 3 + a n z 2 + z f n (z) = z 3 + a n z 2 + z and the sequence (F n) (F n) of iterates F n = f n ∘ ⋯ ∘ f 1 F n = f n ∘ ⋯ ∘ f 1. The Fatou set F 0 F 0 is the set of all z ∈ C ^ z ∈ C ^ such that the sequence (F n) (F n) is normal. The complement of the Fatou set is called the Julia set and denoted by J 0 J 0. The aim of this paper is to study some properties of J 0 J 0. As a particular case, when the sequence (a n) (a n) is constant, a n = a a n = a , then the iteration F n F n becomes the classical iteration f n f n where f (z) = z 3 + a z 2 + z f (z) = z 3 + a z 2 + z. The connectedness locus, M M , is the set of all a ∈ C a ∈ C such that the Julia set is connected. In this paper we investigate some symmetric properties of M M as well. [ABSTRACT FROM AUTHOR]

For β > 1 a real algebraic integer (the base), the finite alphabets A ⊂ Z which realize the identity Q (β) = Per A (β) , where Per A (β) is the set of complex numbers which are (β , A) -eventually periodic representations, are investigated. Comparing with the greedy algorithm, minimal and natural alphabets are defined. The natural alphabets are shown to be correlated to the asymptotics of the Pierce numbers of the base β and Lehmer's problem. The notion of rewriting trail is introduced to construct intermediate alphabets associated with small polynomial values of the base. Consequences on the representations of neighbourhoods of the origin in Q (β) , generalizing Schmidt's theorem related to Pisot numbers, are investigated. Applications to Galois conjugation are given for convergent sequences of bases γ s : = γ n , m 1 , ... , m s such that γ s - 1 is the unique root in (0, 1) of an almost Newman polynomial of the type - 1 + x + x n + x m 1 + ⋯ + x m s , n ≥ 3 , s ≥ 1 , m 1 - n ≥ n - 1 , m q + 1 - m q ≥ n - 1 for all q ≥ 1 . For β > 1 a reciprocal algebraic integer close to one, the poles of modulus < 1 of the dynamical zeta function of the β -shift ζ β (z) are shown, under some assumptions, to be zeroes of the minimal polynomial of β . [ABSTRACT FROM AUTHOR]

We study {0, 1} and {−1, 1} polynomials ƒ(z), called Newman and Littlewood polynomials, that have a prescribed number N(ƒ) of zeros in the open unit disk D = {z ∈ C: |z| < 1}. For every pair (k, n) ∈ N2, where n ≥ 7 and k ∈ [3, n−3], we prove that it is possible to find a {0, 1}-polynomial ƒ(z) of degree °{ƒ} = n with non-zero constant term ƒ(0) ≠ 0, such that N(ƒ) = k and ƒ(z) ≠ 0 on the unit circle ∂ D. On the way to this goal, we answer a question of D. W. Boyd from 1986 on the smallest degree Newman polynomial that satisfies |ƒ(z)| > 2 on the unit circle ∂ D. This polynomial is of degree 38 and we use this special polynomial in our constructions. We also identify (without a proof) all exceptional (k, n) with k ∈ {1, 2, 3, n−3, n−2, n−1}, for which no such {0, 1}-polynomial of degree n exists: such pairs are related to regular (real and complex) Pisot numbers. Similar, but less complete results for {−1, 1} polynomials are established. We also look at the products of spaced Newman polynomials and consider the rotated large Littlewood polynomials. Lastly, based on our data, we formulate a natural conjecture about the statistical distribution of N(ƒ) in the set of Newman and Littlewood polynomials. [ABSTRACT FROM AUTHOR]

Given any n-tuple of complex numbers, one can easily define a canonical polynomial of degree n + 1 that has the entries of this n-tuple as its critical points. In 2002, Beardon, Carne, and Ng studied a map θ : Cn → Cn which outputs the critical values of the canonical polynomial constructed from the input, and they proved that this map is onto. Along the way, they showed that ⋸ is a local homeomorphism whenever the entries of the input are distinct and nonzero, and, implicitly, they produced a polynomial expression for the Jacobian determinant of θ. In this article we extend and generalize both the local homeomorphism result and the elegant determinant identity to analogous situations where the critical points occur with multiplicities. This involves stratifying Cn according to which coordinates are equal and generalizing θ to a similar map Cl → Cl where l is the number of distinct critical points. The more complicated determinant identity that we establish is closely connected to the multinomial identity known as Dyson's conjecture. [ABSTRACT FROM AUTHOR]

We propose general notions to deal with large scale polynomial optimization problems and demonstrate their efficiency on a key industrial problem of the 21st century, namely the optimal power ow problem. These notions enable us to find global minimizers on instances with up to 4,500 variables and 14,500 constraints. First, we generalize the Lasserre hierarchy from real to complex numbers in order to enhance its tractability when dealing with complex polynomial optimization. Complex numbers are typically used to represent oscillatory phenomena, which are omnipresent in physical systems. Using the notion of hyponormality in operator theory, we provide a finite convergence criterion which generalizes the Curto{Fialkow conditions of the real Lasserre hierarchy. Second, we introduce the multi-ordered Lasserre hierarchy in order to exploit sparsity in polynomial optimization problems (in real or complex variables) while preserving global convergence. It is based on two ideas: (1) to use a different relaxation order for each constraint, and (2) to iteratively seek a closest measure to the truncated moment data until a measure matches the truncated data. Third and last, we exhibit a block diagonal structure of the Lasserre hierarchy in the presence of commonly encountered symmetries. To the best of our knowledge, the Lasserre hierarchy was previously limited to small scale problems, while we solve a large scale industrial problem with thousands of variables and constraints to global optimality. [ABSTRACT FROM AUTHOR]