Factorization Of Polynomials Research Articles

Reformulation of Hensel’s Lemma and extension of a theorem of Ore

In this paper, we extend the theorem of Ore regarding factorization of polynomials over p-adic numbers to henselian valued fields of arbitrary rank thereby generalizing the main results of Khanduja and Kumar (J Pure Appl Algebra 216:2648–2656, 2012) and Cohen et al. (Mathematika 47:173–196, 2000). As an application, we derive the analogue of Dedekind’s Theorem regarding splitting of rational primes in algebraic number fields as well as of its converse for general valued fields extending similar results proved for discrete valued fields in Khanduja and Kumar (Int J Number Theory 4:1019–1025, 2008). The generalized version of Ore’s Theorem leads to an extension of a result of Weintraub dealing with a generalization of Eisenstein Irreducibility Criterion (cf. Weintraub in Proc Am Math Soc 141:1159–1160, 2013). We also give a reformulation of Hensel’s Lemma for polynomials with coefficients in henselian valued fields which is used in the proof of the extended Ore’s Theorem and was proved in Khanduja and Kumar (J Algebra Appl 12:1250125, 2013) in the particular case of complete rank one valued fields.

A Note on the $\gamma$-Coefficients of the Tree Eulerian Polynomial

We consider the generating polynomial of the number of rooted trees on the set $\{1,2,\dots,n\}$ counted by the number of descending edges (a parent with a greater label than a child). This polynomial is an extension of the descent generating polynomial of the set of permutations of a totally ordered $n$-set, known as the Eulerian polynomial. We show how this extension shares some of the properties of the classical one. A classical product formula shows that this polynomial factors completely over the integers. From this product formula it can be concluded that this polynomial has positive coefficients in the $\gamma$-basis and we show that a formula for these coefficients can also be derived. We discuss various combinatorial interpretations of these coefficients in terms of leaf-labeled binary trees and in terms of the Stirling permutations introduced by Gessel and Stanley. These interpretations are derived from previous results of Liu, Dotsenko-Khoroshkin, Bershtein-Dotsenko-Khoroshkin, González D'León-Wachs and Gonzláez D'León related to the free multibracketed Lie algebra and the poset of weighted partitions.

A family of tests for irreducibility of polynomials

We combine a simple observation about factorization of polynomials with some previous work to obtain a family of irreducibility tests, which we then illustrate.

T -sion of two Polynomial Sequences and Factorization Properties

<i>t</i> -sion of two Polynomial Sequences and Factorization Properties

Trisection for genus 2 curves in odd characteristic

We provide trisection (division by 3) algorithms for Jacobians of genus 2 curves over finite fields $$\mathbb {F}_q$$Fq of odd characteristic which rely on the factorization of a polynomial whose roots correspond (bijectively) to the set of trisections of the given divisor. We also construct a polynomial whose roots allow us to calculate the 3-torsion divisors. We show the relation between the rank of the 3-torsion subgroup and the factorization of this 3-torsion polynomial, and describe the factorization of the trisection polynomials in terms of the Galois structure of the 3-torsion subgroup. We also generalize these ideas for $$\ell \in \{5,7\}$$lź{5,7}.

Antenna Array Design in MIMO Radar Using NSK Polynomial Factorization Algorithm

The work presented here is concerned with the antenna array design in collocated multiple-input multiple-output (MIMO) radars. After knowing the system requirements, the antenna array design problem is formulated as a standard polynomial factorization. In addition, an algorithm based on Newton-Schubert-Kronecker (NSK) polynomial factorization is proposed. The algorithm contains three steps. First, linear factors are extracted by extended Vieta theorem. Then, undermined high-order factors are confirmed with Newton interpolation and certain high-order factors should be searched for within the undermined ones. Finally, the antenna array configurations are determined according to the result of polynomial factorization. Simulations confirm the wide use of the proposed algorithm in MIMO radar antenna array design.

Computing the Mazur and Swinnerton-Dyer critical subgroup of elliptic curves

Let $E$ be an optimal elliptic curve defined over $\mathbb{Q}$. The critical subgroup of $E$ is defined by Mazur and Swinnerton-Dyer as the subgroup of $E(\mathbb{Q})$ generated by traces of branch points under a modular parametrization of $E$. We prove that for all rank two elliptic curves with conductor smaller than 1000, the critical subgroup is torsion. First, we define a family of critical polynomials attached to $E$ and describe two algorithms to compute such polynomials. We then give a sufficient condition for the critical subgroup to be torsion in terms of the factorization of critical polynomials. Finally, a table of critical polynomials is obtained for all elliptic curves of rank two and conductor smaller than 1000, from which we deduce our result.

Spatial Straight-Line Linkages by Factorization of Motion Polynomials

We use the recently introduced factorization of motion polynomials for constructing overconstrained spatial linkages with a straight-line trajectory. Unlike previous examples of such linkages by other authors, they are single-loop linkages and the end-effector motion is not translational. In particular, we obtain a number of linkages with four revolute and two prismatic joints and a remarkable linkage with seven revolute joints one of whose joints performs a Darboux motion.

Deterministic root finding in finite fields

Finding roots of polynomials with coefficients in a finite field is a special instance of the polynomial factorization problem. The most well known algorithm for factoring and root-finding is the Cantor-Zassenhaus algorithm. It is a Las Vegas algorithm with running time polynomial in both the polynomial degree and the field size. Its running time is quasi-optimal for the special case of root-finding. No deterministic polynomial time algorithm for these problems is known in general, however several deterministic algorithms are known for special instances, most notably when the characteristic of the finite field is small. The goal of this poster is to review the best deterministic algorithms for root-finding, in a systematic way. We present, in a unified way, four algorithms: • Berlekamp's Trace Algorithm [2] (BTA), • Moenk's Subgroup Refinement Method [7] (SRM), • Menezes, van Oorschot and Vanstone's Affine Refinement Method [5, 10] (ARM), and • Petit's Successive Resultants Algorithm [8] (SRA). It is the first time that these algorithms are presented together in a comprehensive way, and that they are rigorously analysed, implemented and compared to each other. In doing so, we obtain several new results: • We significantly improve the complexity ARM, matching that of BTA and SRA. • We highlight a profound duality relationship between ARM and SRA. • We show how to combine ARM with SRM to obtain a new algorithm, which always performs better, and of which ARM and SRM are special instances. The new algorithm considerably extends the range of finite fields for which deterministic polynomial time root-finding is possible. • We present several practical and asymptotic improvements to special instances of the algorithms. Part of these results were submitted in response to the call for papers of ISSAC '15, but were rejected. This poster corrects some minor imperfections, improves the asymptotic complexities of some algorithms, and presents a new algorithm not previously known.

The Numerical Factorization of Polynomials

Polynomial factorization in conventional sense is an ill-posed problem due to its discontinuity with respect to coefficient perturbations, making it a challenge for numerical computation using empirical data. As a regularization, this paper formulates the notion of numerical factorization based on the geometry of polynomial spaces and the stratification of factorization manifolds. Furthermore, this paper establishes the existence, uniqueness, Lipschitz continuity, condition number, and convergence of the numerical factorization to the underlying exact factorization, leading to a robust and efficient algorithm with a MATLAB implementation capable of accurate polynomial factorizations using floating point arithmetic even if the coefficients are perturbed.

Bounded-degree factors of lacunary multivariate polynomials

In this paper, we present a new method for computing bounded-degree factors of lacunary multivariate polynomials. In particular for polynomials over number fields, we give a new algorithm that takes as input a multivariate polynomial f in lacunary representation and a degree bound d and computes the irreducible factors of degree at most d of f in time polynomial in the lacunary size of f and in d. Our algorithm, which is valid for any field of zero characteristic, is based on a new gap theorem that enables reducing the problem to several instances of (a) the univariate case and (b) low-degree multivariate factorization.The reduction algorithms we propose are elementary in that they only manipulate the exponent vectors of the input polynomial. The proof of correctness and the complexity bounds rely on the Newton polytope of the polynomial, where the underlying valued field consists of Puiseux series in a single variable.

A polynomial chaos ensemble hydrologic prediction system for efficient parameter inference and robust uncertainty assessment

Summary This paper presents a polynomial chaos ensemble hydrologic prediction system (PCEHPS) for an efficient and robust uncertainty assessment of model parameters and predictions, in which possibilistic reasoning is infused into probabilistic parameter inference with simultaneous consideration of randomness and fuzziness. The PCEHPS is developed through a two-stage factorial polynomial chaos expansion (PCE) framework, which consists of an ensemble of PCEs to approximate the behavior of the hydrologic model, significantly speeding up the exhaustive sampling of the parameter space. Multiple hypothesis testing is then conducted to construct an ensemble of reduced-dimensionality PCEs with only the most influential terms, which is meaningful for achieving uncertainty reduction and further acceleration of parameter inference. The PCEHPS is applied to the Xiangxi River watershed in China to demonstrate its validity and applicability. A detailed comparison between the HYMOD hydrologic model, the ensemble of PCEs, and the ensemble of reduced PCEs is performed in terms of accuracy and efficiency. Results reveal temporal and spatial variations in parameter sensitivities due to the dynamic behavior of hydrologic systems, and the effects (magnitude and direction) of parametric interactions depending on different hydrological metrics. The case study demonstrates that the PCEHPS is capable not only of capturing both expert knowledge and probabilistic information in the calibration process, but also of implementing an acceleration of more than 10 times faster than the hydrologic model without compromising the predictive accuracy.

Sublinear separators, fragility and subexponential expansion

Let G be a subgraph-closed graph class with bounded maximum degree. We show that if G has balanced separators whose size is smaller than linear by a polynomial factor, then G has subexponential expansion. This gives a partial converse to a result of Nešetřil and Ossona de Mendez. As an intermediate step, the proof uses a new kind of graph decompositions.

COMPLEX FACTORIZATIONS OF THE GENERALIZED FIBONACCI SEQUENCES {qn}

In this note, we consider a generalized Fibonacci sequence $\{q_n\}$. Then give a connection between the sequence $\{q_n\}$ and the Chebyshev polynomials of the second kind $U_n(x)$. With the aid of factorization of Chebyshev polynomials of the second kind $U_n(x)$, we derive the complex factorizations of the sequence $\{q_n\}$.

The Parameterized Complexity of Geometric Graph Isomorphism

We study the parameterized complexity of Geometric Graph Isomorphism (Known as the Point Set Congruence problem in computational geometry): given two sets of n points A and B with rational coordinates in k-dimensional euclidean space, with k as the fixed parameter, the problem is to decide if there is a bijection $$\pi :A \rightarrow B$$ź:AźB such that for all $$x,y \in A$$x,yźA, $$\Vert x-y\Vert = \Vert \pi (x)-\pi (y)\Vert $$źx-yź=źź(x)-ź(y)ź, where $$\Vert \cdot \Vert $$ź·ź is the euclidean norm. Our main result is the following: We give a $$O^*(k^{O(k)})$$Oź(kO(k)) time (The $$O^*(\cdot )$$Oź(·) notation here, as usual, suppresses polynomial factors) FPT algorithm for Geometric Isomorphism. This is substantially faster than the previous best time bound of $$O^*(2^{O(k^4)})$$Oź(2O(k4)) for the problem (Evdokimov and Ponomarenko in Pure Appl Algebra 117---118:253---276, 1997). In fact, we show the stronger result that even canonical forms for finite point sets with rational coordinates can also be computed in $$O^*(k^{O(k)})$$Oź(kO(k)) time. We also briefly discuss the isomorphism problem for other $$l_p$$lp metrics. Specifically, we describe a deterministic polynomial-time algorithm for finite point sets in $$\mathbb {Q}^2$$Q2.

Factoring the characteristic polynomial of a lattice

We introduce a new method for showing that the roots of the characteristic polynomial of certain finite lattices are all nonnegative integers. This method is based on the notion of a quotient of a poset which will be developed to explain this factorization. Our main theorem will give two simple conditions under which the characteristic polynomial factors with nonnegative integer roots. We will see that Stanley's Supersolvability Theorem is a corollary of this result. Additionally, we will prove a theorem which gives three conditions equivalent to factorization. To our knowledge, all other theorems in this area only give conditions which imply factorization. This theorem will be used to connect the generating function for increasing spanning forests of a graph to its chromatic polynomial. We finish by mentioning some other applications of quotients of posets as well as some open questions.

The factorization of the linear differential operator

Abstract In this paper, necessary and sufficient conditions for factorization of a linear differential operator are presented. As a consequence of the factorization result some criterion of polynomial factorization is obtained. As a special case of the main result we have got Polynomial Remainder Theorem.

Assigning Channels Via the Meet-in-the-Middle Approach

We study the complexity of the Channel Assignment problem. By applying the meet-in-the-middle approach we get an algorithm for the $$\ell $$l-bounded Channel Assignment (when the edge weights are bounded by $$\ell $$l) running in time $$O^*((2\sqrt{\ell +1})^n)$$O?((2l+1)n). This is the first algorithm which breaks the $$(O(\ell ))^n$$(O(l))n barrier. We extend this algorithm to the counting variant, at the cost of slightly higher polynomial factor. Very recently the second author showed that Channel Assignment does not admit a $$O(c^n)$$O(cn)-time algorithm, for a constant c independent of $$\ell $$l. We consider a similar question for Generalized $$T$$T-Coloring, a CSP problem that generalizes Channel Assignment. We show that Generalized $$T$$T-Coloring does not admit a $$2^{2^{o\left( \sqrt{n}\right) }} \mathrm{poly}(r)$$22onpoly(r)-time algorithm, where r is the size of the instance.

Pebbling, Entropy, and Branching Program Size Lower Bounds

We contribute to the program of proving lower bounds on the size of branching programs solving the Tree Evaluation Problem introduced by Cook et al. [2012]. Proving a superpolynomial lower bound for the size of nondeterministic thrifty branching programs would be an important step toward separating NL from P using the tree evaluation problem. First, we show that Read-Once Nondeterministic Thrifty BPs are equivalent to whole black-white pebbling algorithms, thus showing a tight lower bound (ignoring polynomial factors) for this model. We then introduce a weaker restriction of nondeterministic thrifty branching programs called Bitwise Independence . The best known [Cook et al. 2012] nondeterministic thrifty branching programs (of size O ( k h /2 + 1 )) for the tree evaluation problem are Bitwise Independent. As our main result, we show that any Bitwise Independent Nondeterministic Thrifty Branching Program solving BT 2 (h, k) must have at least (k2) h/2 states. Prior to this work, lower bounds were known for nondeterministic thrifty branching programs only for fixed heights h = 2, 3, 4 [Cook et al. 2012]. We prove our results by associating a fractional black-white pebbling strategy with any bitwise independent nondeterministic thrifty branching program solving the Tree Evaluation Problem. Such a connection was not known previously, even for fixed heights. Our main technique is the entropy method introduced by Jukna and Zák [2001] originally in the context of proving lower bounds for read-once branching programs. We also show that the previous lower bounds known [Cook et al. 2012] for deterministic branching programs for the Tree Evaluation Problem can be obtained using this approach. Using this method, we also show tight lower bounds for any k -way deterministic branching program solving the Tree Evaluation Problem when the instances are restricted to have the same group operation in all internal nodes.

Fountain Code Design for the Y-Network

This paper describes a novel approach for designing distributed fountain codes based on polynomial factorization. The proposed approach is used for the well known Y-network configuration. The suggested factorization approach is applicable to LT codes and Raptor codes or any other fountain code describable with a degree distribution. Simulation results verify the success of this technique.