Factorization Of Polynomials Research Articles

Exact Algorithms for the Minimum Load Spanning Tree Problem

In a minimum load spanning tree (MLST) problem, we are given an undirected graph and nondecreasing load functions for nodes defined on nodes’ degrees in a spanning tree, and the objective is to find a spanning tree that minimizes the maximum load among all nodes. We propose the first [Formula: see text] time exact algorithm for the MLST problem, where [Formula: see text] is the number of nodes and [Formula: see text] ignores polynomial factor. The algorithm is obtained by repeatedly querying whether a candidate objective value is feasible, where each query can be formulated as a bounded degree spanning tree problem (BDST). We propose a novel solution to BDST by extending an inclusion-exclusion based algorithm. To further enhance the time efficiency of the previous algorithm, we then propose a faster algorithm by generalizing the concept of branching walks. In addition, for the purpose of comparison, we give the first mixed integer linear programming formulation for MLST. In numerical analysis, we consider various load functions on a randomly generated network. The results verify the effectiveness of the proposed algorithms. Summary of Contribution: Minimum load spanning tree (MLST) plays an important role in various applications such as wireless sensor networks (WSNs). In many applications of WSNs, we often need to collect data from all sensors to some specified sink. In this paper, we propose the first exact algorithms for the MLST problem. Besides having theoretical guarantees, our algorithms have extraordinarily good performance in practice. We believe that our results make significant contributions to the field of graph theory, internet of things, and WSNs.

STRICTLY REAL FUNDAMENTAL THEOREM OF ALGEBRA USING POLYNOMIAL INTERLACING

AbstractWithout resorting to complex numbers or any advanced topological arguments, we show that any real polynomial of degree greater than two always has a real quadratic polynomial factor, which is equivalent to the fundamental theorem of algebra. The proof uses interlacing of bivariate polynomials similar to Gauss’s first proof of the fundamental theorem of algebra using complex numbers, but in a different context of division residues of strictly real polynomials. This shows the sufficiency of basic real analysis as the minimal platform to prove the fundamental theorem of algebra.

Iterative Alpha Expansion for Estimating Gradient-Sparse Signals from Linear Measurements

AbstractWe consider estimating a piecewise-constant image, or a gradient-sparse signal on a general graph, from noisy linear measurements. We propose and study an iterative algorithm to minimize a penalized least-squares objective, with a penalty given by the “ℓ0-norm” of the signal’s discrete graph gradient. The method uses a non-convex variant of proximal gradient descent, applying the alpha-expansion procedure to approximate the proximal mapping in each iteration, and using a geometric decay of the penalty parameter across iterations to ensure convergence. Under a cut-restricted isometry property for the measurement design, we prove global recovery guarantees for the estimated signal. For standard Gaussian designs, the required number of measurements is independent of the graph structure, and improves upon worst-case guarantees for total-variation (TV) compressed sensing on the 1-D line and 2-D lattice graphs by polynomial and logarithmic factors respectively. The method empirically yields lower mean-squared recovery error compared with TV regularization in regimes of moderate undersampling and moderate to high signal-to-noise, for several examples of changepoint signals and gradient-sparse phantom images.

Computation of residual polynomial operators of inductive valuations

Let ( K , v ) be a valued field, and μ an inductive valuation on K [ x ] extending v . Let G μ be the graded algebra of μ over K [ x ] , and κ the maximal subfield of the subring of G μ formed by the homogeneous elements of degree zero. In this paper, we find an algorithm to compute the field κ and the residual polynomial operator R μ : K [ x ] → κ [ y ] , where y is another indeterminate, without any need to perform computations in the graded algebra. This leads to an OM algorithm to compute the factorization of separable defectless polynomials over henselian fields.

Characteristic polynomial and eigenvalues of antiadjacency matrix of directed unicyclic flower vase graph

This research discussed the characteristic polynomial and eigenvalues of antiadjacency matrix of directed unicyclic flower vase graph. The entries of the antiadjacency matrix of a directed graph represent the presence or the absence of a directed arc from one vertex to the others. If A is the adjacency matrix of a graph , then the antiadjacency matrix B of graph is B = J − A, where J is the square matrix with all entries equal to one. The general form of characteristic polynomial coefficients of the antiadjacency matrix of directed unicyclic flower vase graph can be obtained by calculating the sum of determinants of the antiadjacency matrices of all induced cyclic and acyclic subgraphs, while the eigenvalues were obtained by using polynomial factorization and Horner’s method. This paper gives the characteristic polynomial coefficients and eigenvalues of antiadjacency matrix of directed unicyclic flower vase graph. The characteristic polynomial can be considered as a function that depends on the number of vertices.

Properties of characteristic polynomial and eigenvalues of antiadjacency matrix of directed unicyclic helm graph

A directed unicyclic graph is a directed graph that has only one directed cycle subgraph. A directed unicyclic helm graph is obtained from a directed wheel graph by adjoining a directed pendant edge at each vertex of the cycle. A directed graph can be represented into several matrix representations, one of them is the antiadjacency matrix. The antiadjacency matrix is a matrix in which the entries represent whether there is a directed edge from one vertex to another. This paper discusses the general form of the coefficients of the characteristic polynomial that obtained by adding all of the determinants of antiadjacency matrix from each induced acyclic and cyclic subgraphs. The eigenvalues of the antiadjacency matrix of the directed unicyclic helm graph obtained by polynomial factorization. The result obtained denotes that the coefficients of the characteristic polynomial and eigenvalues of the antiadjacency matrix depend on the number of vertices of the cycle subgraphs of directed unicyclic helm graph.

On a class of factors of the Chebyshev polynomials

On a class of factors of the Chebyshev polynomials

On a Fejér–Riesz factorization of generalized trigonometric polynomials

Function theory on the unit disc proved key to a range of problems in statistics, probability theory, signal processing literature, and applications, and in this, a special place is occupied by trigonometric functions and the Fejer-Riesz theorem that non-negative trigonometric polynomials can be expressed as the modulus of a polynomial of the same degree evaluated on the unit circle. In the present note we consider a natural generalization of non-negative trigonometric polynomials that are matrix-valued with specified non-trivial poles (i.e., other than at the origin or at infinity). We are interested in the corresponding spectral factors and, specifically, we show that the factorization of trigonometric polynomials can be carried out in complete analogy with the Fejer-Riesz theorem. The affinity of the factorization with the Fejer-Riesz theorem and the contrast to classical spectral factorization lies in the fact that the spectral factors have degree smaller than what standard construction in factorization theory would suggest. We provide two juxtaposed proofs of this fundamental theorem, albeit for the case of strict positivity, one that relies on analytic interpolation theory and another that utilizes classical factorization theory based on the Yacubovich-Popov-Kalman (YPK) positive-real lemma.

The irreducibility of some Wronskian Hermite polynomials

We study the irreducibility in Z[x] of Wronskian Hermite polynomials labelled by partitions. It is known that these polynomials factor as a power of x times a remainder polynomial. We show that the remainder polynomial is irreducible for the partitions (n,m) with m≤2, and (n,n) when n+1 is a square.Our main tools are two theorems that we prove for all partitions. The first result gives a sharp upper bound for the slope of the edges of the Newton polygon for the remainder polynomial. The second result is a Schur-type congruence for Wronskian Hermite polynomials.We also explain how irreducibility determines the number of real zeros of Wronskian Hermite polynomials, and prove Veselov’s conjecture for partitions of the form (n,k,k−1,…,1).

Shifting powers in Spivey’s Bell number formula

In this paper, we consider extensions of Spivey’s Bell number formula wherein the argument of the polynomial factor is translated by an arbitrary amount. This idea is applied more generally to the r-Whitney numbers of the second kind, denoted by W (n, k), where some new identities are found by means of algebraic and combinatorial arguments. The former makes use of infinite series manipulations and Dobinski-like formulas satisfied by W (n, k), whereas the latter considers distributions of certain statistics on the underlying enumerated class of set partitions. Further-more, these two approaches provide new ways in which to deduce the Spivey formula for W (n, k). Finally, we establish an analogous result involving the r-Lah numbers wherein the order matters in which the elements are written within the blocks of the aforementioned set partitions.

On the basic properties of GCn sets

In this paper the simplest [Formula: see text]-correct sets in the plane — [Formula: see text] sets are studied. An [Formula: see text]-correct node set [Formula: see text] is called [Formula: see text] set if the fundamental polynomial of each node is a product of [Formula: see text] linear factors. We say that a node uses a line if the line is a factor of the fundamental polynomial of this node. A line is called [Formula: see text]-node line if it passes through exactly [Formula: see text] nodes of [Formula: see text] At most [Formula: see text] nodes can be collinear in any [Formula: see text]-correct set and an [Formula: see text]-node line is called a maximal line. The Gasca–Maeztu conjecture (1982) states that for every [Formula: see text] set there exists at least one maximal line. Until now the conjecture has been proved only for the cases [Formula: see text] Here, for a line [Formula: see text] we introduce and study the concept of [Formula: see text]-lowering of the set [Formula: see text] and define so called proper lines. We also provide refinements of several basic properties of [Formula: see text] sets regarding the maximal lines, [Formula: see text]-node lines, the used lines, as well as the subset of nodes that use a given line.

Nearly irreducibility of polynomials and the Newton diagrams

Abstract Let f be a polynomial in two complex variables. We say that f is nearly irreducible if any two nonconstant polynomial factors of f have a common zero in C2. In the paper we give a criterion of nearly irreducibility for a given polynomial f in terms of its Newton diagram.

New Method of Solving the Economic Complex Systems

In this study, the authors first develop a direct method used to solve the linear nonhomogeneous time-invariant difference equation with the same number for inputs and outputs. Economic cybernetics is the crystallization for the integration of economics and cybernetics. It analyzes the stability, controllability, and observability of the economic system by establishing a system model and enables people to better understand the characteristics of the economic system and solve economic optimization problems. The economic model generally applies the discrete recurrence difference equation. The significant analytic approach for the difference equation is the z-transformation technique. The z-transformation state of the economic cybernetics state-space difference equation generally is a rational function with the same power for the numerator and the denominator. The proposed approach will take the place of the traditional methods without all annoying procedures involving the long division of some complicated polynomials, the expanded multiplication of many polynomial factors, the differentiation of some complicated polynomials, and the complex derivations of all partial fraction parameters. To highlight the novelty of this research, this study especially applies the proposed theorems originally belonging to engineering to the field of economic applications.

The rank of Pseudo walk matrices: controllable and recalcitrant pairs

A pseudo walk matrix \(\mathbf{W}_\mathbf{v}\) of a graph \(G\) having adjacency matrix \(\mathbf{A}\) is an \(n\times n\) matrix with columns \(\mathbf{v},\mathbf{A}\mathbf{v},\mathbf{A}^2\mathbf{v},\ldots,\mathbf{A}^{n-1}\mathbf{v}\) whose Gram matrix has constant skew diagonals, each containing walk enumerations in \(G\). We consider the factorization over \(\mathbb{Q}\) of the minimal polynomial \(m(G,x)\) of \(\mathbf{A}\). We prove that the rank of \(\mathbf{W}_\mathbf{v}\), for any walk vector \(\mathbf{v}\), is equal to the sum of the degrees of some, or all, of the polynomial factors of \(m(G,x)\). For some adjacency matrix \(\mathbf{A}\) and a walk vector \(\mathbf{v}\), the pair \((\mathbf{A},\mathbf{v})\) is controllable if \(\mathbf{W}_\mathbf{v}\) has full rank. We show that for graphs having an irreducible characteristic polynomial over \(\mathbb{Q}\), the pair \((\mathbf{A},\mathbf{v})\) is controllable for any walk vector \(\mathbf{v}\). We obtain the number of such graphs on up to ten vertices, revealing that they appear to be commonplace. It is also shown that, for all walk vectors \(\mathbf{v}\), the degree of the minimal polynomial of the largest eigenvalue of \(\mathbf{A}\) is a lower bound for the rank of \(\mathbf{W}_\mathbf{v}\). If the rank of \(\mathbf{W}_\mathbf{v}\) attains this lower bound, then \((\mathbf{A},\mathbf{v})\) is called a recalcitrant pair. We reveal results on recalcitrant pairs and present a graph having the property that \((\mathbf{A},\mathbf{v})\) is neither controllable nor recalcitrant for any walk vector \(\mathbf{v}\).

Algorithms for CRT-variant of Approximate Greatest Common Divisor Problem

AbstractThe approximate greatest common divisor problem (ACD) and its variants have been used to construct many cryptographic primitives. In particular, the variants of the ACD problem based on Chinese remainder theorem (CRT) are being used in the constructions of a batch fully homomorphic encryption to encrypt multiple messages in one ciphertext. Despite the utility of the CRT-variant scheme, the algorithms that secures its security foundation have not been probed well enough.In this paper, we propose two algorithms and the results of experiments in which the proposed algorithms were used to solve the variant problem. Both algorithms take the same time complexity $\begin{array}{} \displaystyle 2^{\tilde{O}(\frac{\gamma}{(\eta-\rho)^2})} \end{array}$ up to a polynomial factor to solve the variant problem for the bit size of samples γ, secret primes η, and error bound ρ. Our algorithm gives the first parameter condition related to η and γ size. From the results of the experiments, it has been proved that the proposed algorithms work well both in theoretical and experimental terms.

On the rational relationships among pseudo-roots of a non-commutative polynomial

For a non-commutative ring R , we consider factorizations of polynomials in R [ t ] where t is a central variable. A pseudo-root of a polynomial p ( t ) = p 0 + p 1 t + ⋯ p k t k is an element ξ ∈ R , for which there exist polynomials q 1 , q 2 such that p = q 1 ( t − ξ ) q 2 . We investigate the rational relationships that hold among the pseudo-roots of p ( t ) by using the diamond operations for cover graphs of modular lattices . When R is a division ring, each finite subset S of R corresponds to a unique minimal monic polynomial f S that vanishes on S . By results of Leroy and Lam [16] , the set of polynomials { f T : T ⊆ S } with the right-divisibility order forms a lattice with join operation corresponding to (left) least common multiple and meet operation corresponding to (right) greatest common divisor. The set of edges of the cover graph of this lattice correspond naturally to a set Λ S of pseudo-roots of f S . Given an arbitrary subset of Λ S , our results provide a graph theoretic criterion that guarantees that the subset rationally generates all of Λ S , and in particular, rationally generates S .

Efficiently factoring polynomials modulo p4

Polynomial factoring has famous practical algorithms over fields– finite, rational and p-adic. However, modulo prime powers, factoring gets harder because there is non-unique factorization and a combinatorial blowup ensues. For example, x2+pmodp2 is irreducible, but x2+pxmodp2 has exponentially many factors in the input size (which here is logarithmic in p)! We present the first randomized poly(deg⁡f,log⁡p) time algorithm to factor a given univariate integral polynomial f modulo pk, for a prime p and k≤4.1 Thus, we solve the open question of factoring modulo p3 posed in (Sircana, ISSAC'17).Our method reduces the general problem of factoring fmodpk to that of root finding of a related polynomial E(y)mod〈pk,φ(x)ℓ〉 for some irreducible φmodp. We can efficiently solve the latter for k≤4, by incrementally transforming E. Moreover, we discover an efficient refinement of Hensel lifting to lift factors of fmodp to those modp4 (if possible). This was previously unknown, as the case of repeated factors of fmodp forbids classical Hensel lifting.

Analytic expressions for the positive definite and unimodal regions of Gram-Charlier series

It often arises in practice that, although the first few moments of a distribution are known, the density of the distribution cannot be determined in closed form. In such cases, Gram-Charlier and Edgeworth series are commonly used to analytically approximate the unknown density in terms of the known moments. Although convenient, these series contain polynomial factors, and can hence lead to density approximations taking negative values or becoming multimodal in general. Consequently it is of interest to determine the set of moments for which the corresponding density approximations are positive definite and unimodal. In contrast to the existing literature that determines the boundaries of such sets numerically, explicit analytic expressions for the two boundaries are given in this paper for the Gram-Charlier series. Moreover, a method for projecting a given set of moments onto the boundaries of the two regions in order to minimizes the Kolmogorov-Smirnov statistic of corresponding density approximations is also provided.

Correction To: The Complexity of Factors of Multivariate Polynomials

Vladimir Lysikov kindly pointed out an error in the proof of Theorem 5.7. We provide here a corrected statement and its proof.

Exact results and Schur expansions in quiver Chern-Simons-matter theories

We study several quiver Chern-Simons-matter theories on the three-sphere, combining the matrix model formulation with a systematic use of Mordell’s integral, computing partition functions and checking dualities. We also consider Wilson loops in ABJ(M) theories, distinguishing between typical (long) and atypical (short) representations and focusing on the former. Using the Berele-Regev factorization of supersymmetric Schur polynomials, we express the expectation value of the Wilson loops in terms of sums of observables of two factorized copies of U(N ) pure Chern-Simons theory on the sphere. Then, we use the Cauchy identity to study the partition functions of a number of quiver Chern-Simons-matter models and the result is interpreted as a perturbative expansion in the parameters tj = −e2πmj , where mj are the masses. Through the paper, we incorporate different generalizations, such as deformations by real masses and/or Fayet-Iliopoulos parameters, the consideration of a Romans mass in the gravity dual, and adjoint matter.