Factorization Of Polynomials Research Articles

Why the characteristic polynomial factors

We survey three methods for proving that the characteristic polynomial of a finite ranked lattice factors over the nonnegative integers and indicate how they have evolved recently. The first technique uses geometric ideas and is based on Zaslavsky’s theory of signed graphs. The second approach is algebraic and employs results of Saito and Terao about free hyperplane arrangements. Finally we consider a purely combinatorial theorem of Stanley about supersolvable lattices and its generalizations.

An introductory course in commutative algebra

1: Rings. 2: Euclidean rings. 3: Highest common factor. 4: The four-squares theorem. 5: Fields and polynomials. 6: Unique factorization domains. 7: Field of quotients of an integral domain. 8: Factorization of polynomials. 9: Fields and field extensions. 10: Finite cyclic groups and finite fields. 11: Algebraic numbers. 12: Ruler and Compass constructions. 13: Homomorphisms, ideals and factor rings. 14: Principal ideal domains and a method for constructing fields. 15: Finite fields. Solutions to selected exercises. References

On the degrees of irreducible factors of polynomials over a finite field

Let F q [X] denote the multiplicative semigroups of monic polynomials in one indeterminate X, over a finite field F q. We determine for each fixed q, the asymptotic average number of distinct degrees of the irreducible factors of a polynomial of degree n in F q [X]. The results are of relevance to various polynomial factorization algorithms.

The complexity of theory revision

A knowledge-based system uses its database (also known as its “theory”) to produce answers to the queries it receives. Unfortunately, these answers may be incorrect if the underlying theory is faulty. Standard “theory revision” systems use a given set of “labeled queries” (each a query paired with its correct answer) to transform the given theory, by adding and/or deleting either rules and/or antecedents, into a related theory that is as accurate as possible. After formally defining the theory revision task, this paper provides both sample and computational complexity bounds for this process. It first specifies the number of labeled queries necessary to identify a revised theory whose error is close to minimal with high probability. It then considers the computational complexity of finding this best theory, and proves that, unless P = NP, no polynomial-time algorithm can identify this optimal revision, even given the exact distribution of queries, except in certain simple situations. It also shows that, except in such simple situations, no polynomial-time algorithm can produce a theory whose error is even close to (i.e., within a particular polynomial factor of) optimal. The first (sample complexity) results suggest reasons why theory revision can be more effective than learning from scratch, while the second (computational complexity) results explain many aspects of the standard theory revision systems, including the practice of hill-climbing to a locally-optimal theory, based on a given set of labeled queries.

Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer

A digital computer is generally believed to be an efficient universal computing device; that is, it is believed to be able to simulate any physical computing device with an increase in computation time by at most a polynomial factor. This may not be true when quantum mechanics is taken into consideration. This paper considers factoring integers and finding discrete logarithms, two problems that are generally thought to be hard on classical computers and that have been used as the basis of several proposed cryptosystems. Efficient randomized algorithms are given for these two problems on a hypothetical quantum computer. These algorithms take a number of steps polynomial in the input size, for example, the number of digits of the integer to be factored.

Minimal Degree Coprime Factorization of Rational Matrices

Given a rational matrix G with complex coefficients and a domain $\Gamma$ in the closed complex plane, both arbitrary, we develop a complete theory of coprime factorizations of G over $\Gamma$, with denominators of McMillan degree as small as possible. The main tool is a general pole displacement theorem which gives conditions for an invertible rational matrix to dislocate by multiplication a part of the poles of G. We apply this result to obtain the parametrized class of all coprime factorizations over $\Gamma$ with denominators of minimal McMillan degree nb---the number of poles of G outside $\Gamma$. Specific choices of the parameters and of $\Gamma$ allow us to determine coprime factorizations, as for instance, with polynomial, proper, or stable factors. Further, we consider the case in which the denominator has a certain symmetry, namely it is J all-pass with respect either to the imaginary axis or to the unit circle. We give necessary and sufficient solvability conditions for the problem of coprime factorization with J all-pass denominator of McMillan degree nb and, when a solution exists, we give a construction of the class of coprime factors. When no such solution exists, we discuss the existence of, and give solutions to, coprime factorizations with J all-pass denominators of minimal McMillan degree (>nb). All the developments are carried out in terms of descriptor realizations associated with rational matrices, leading to explicit and computationally efficient formulas.

A Nonstandard Cyclic Reduction Method, Its Variants and Stability

A nonstandard cyclic reduction method is introduced for solving the Poisson equation in rectangular domains. Different ways of solving the arising reduced systems are considered. The partial solution approach leads to the so-called partial solution variant of the cyclic reduction (PSCR) method, while the other variants are obtained by using the matrix rational polynomial factorization technique, including the partial fraction expansions, the fast Fourier transform (FFT) approach, and the combination of Fourier analysis and cyclic reduction (FACR) techniques. Such techniques have originally been considered in the standard cyclic reduction framework. The equivalence of the partial solution and the partial fraction techniques is shown. The computational cost of the considered variants is ${\mathcal O} (N\log N)$ operations, except for the FACR techniques for which it is ${\mathcal O} (N\log\log N)$. The stability estimate for the considered method is constructed, and the stability is demonstrated by numerical experiments.

FFT-based fast polynomial rooting

A fast root finding algorithm based on an FFT implementation is proposed, thus avoiding the need for a computationally heavy polynomial rooting technique that estimates the eigenvalues of a companion matrix. The minimum-phase polynomial factorisation proposed by Oppenheim and Schafer (1989) is first extended to an arbitrary radius factorisation, then used to extract the roots in an iterative manner.

A Parallel Fast Direct Solver for Block Tridiagonal Systems with Separable Matrices of Arbitrary Dimension

A parallel fast direct solution method for linear systems with separable block tridiagonal matrices is considered. Such systems appear, for example, when discretizing the Poisson equation in a rectangular domain using the five-point finite difference scheme or the piecewise linear finite elements on a triangulated, possibly nonuniform rectangular mesh. The method under consideration has the arithmetical complexity ${\mathcal O}(N\log N)$, and it is closely related to the cyclic reduction method, but instead of using the matrix polynomial factorization, the so-called partial solution technique is employed. Hence, in this paper, the method is called the partial solution variant of the cyclic reduction method (PSCR method). The method is presented and analyzed in a general radix-q framework and, based on this analysis, the radix-4 variant is chosen for parallel implementation using the MPI standard. The generalization of the method to the case of arbitrary block dimension is described. The numerical experiments show the sequential efficiency and numerical stability of the PSCR method compared to the well-known BLKTRI implementation of the generalized cyclic reduction method. The good scalability properties of the parallel PSCR method are demonstrated in a distributed-memory Cray T3E-750 computer.

Philippe Flajolet's Research in Analysis of Algorithms and Combinatorics

Philippe Flajolet's research in theoretical computer science spans over more than 20 years. He made lasting contributions to the analysis of algorithms and analytic combinatorics. Among many of his results we mention here some in such diversified topics as enumeration, number theory, formal languages, continued fractions, automatic analysis of algorithms, Mellin transform, digital sums, recurrences, trees, random generation of combinatorial objects, random graphs and mappings, polynomial factorization, communications, codes, graphics, etc. This gives only a small snapshot of his work, and we encourage the reader to visit Flajolet's homepage http://www-rocq.inria.fr/algo/flajolet/index.html for a fuller account. The bibliography was taken from his homepage and may not be totally complete, although we added a few items. Our paper was written without giving any prior notice to Philippe Flajolet. It reflects the view of the authors and any misunderstandings and shortcomings should be put on their account.

Single-particle dynamics in particle storage rings with integrable polynomial factorization maps

A six-dimensional symplectic tracking code based on the concept of integrable polynomial factorization (IPF) maps has been developed for the study of beam dynamics in large storage rings. To examine the accuracy of the IPF maps, comparison studies on beam dynamics between the IPF maps and the original system have been conducted with two sample lattices: a lattice containing one sextupole kick that is otherwise linear and the LHC (Large Hadron Collider) collision lattice. It was found that the errors in the $n\mathrm{th}$-order IPF map scale as the $(n+1)\mathrm{th}$ power of phase-space amplitude. As the order of the map is increased, the difference between the original system and its IPF map can thus be reduced to be as small as desired. A study of phase-space portraits showed that the IPF map precisely reproduces the resonance structure of the original system even in the phase-space region where the system is quite nonlinear. The tracking study for the LHC showed that the IPF map accurately predicts the same dynamic aperture as does element-by-element tracking. Our results suggest that the IPF map is a reliable model for the study of long-term behavior of beam particles in large storage rings.

Polynomial Factorisation and an Application to Regular Directed Graphs

The main theme is the distribution of polynomials of given degree which split into a product of linear factors over a finite field. The work was motivated by the following problem on regular directed graphs. Extending a notion of Chung, Katz has defined a regular directed graph based on thek-algebrak[X]/(f), wherekis the finite field of orderqandfa monic polynomial of degreenoverk. It is shown that the diameter of this graph is at mostn+2 wheneverq≥B(n)=[n(n+2)!]2. This improves on the work of Katz who gave a similar result for square-free polynomialsfwithout specifyingB(n).

Partial Fraction Decomposition in [formula omitted](z) and Simultaneous Newton Iteration for Factorization in [formula omitted][z

The subject of this paper is fast numerical algorithms for factoring univariate polynomials with complex coefficients and for computing partial fraction decompositions (PFDs) of rational functions in C(z). Numerically stable and computationally feasible versions of PFD are specified first for the special case of rational functions with all singularities in the unit disk (the “bounded case”) and then for rational functions with arbitrarily distributed singularities. Two major algorithms for computing PFDs are presented: The first one is an extension of the “splitting circle method” by A. Schönhage (“The Fundamental Theorem of Algebra in Terms of Computational Complexity,” Technical Report, Univ. Tübingen, 1982) for factoring polynomials in C[z] to an algorithm for PFD. The second algorithm is a Newton iteration for simultaneously improving the accuracy of all factors in an approximate factorization of a polynomial resp. all partial fractions of an approximate PFD of a rational function. Algorithmically useful starting value conditions for the Newton algorithm are provided. Three subalgorithms are of independent interest. They compute the product of a sequence of polynomials, the sum of a sequence of rational functions, and the modular representation of a polynomial. All algorithms are described in great detail, and numerous technical auxiliaries are provided which are also useful for the design and analysis of other algorithms in computational complex analysis. Combining the splitting circle method with simultaneous Newton iteration yields favourable time bounds (measured in bit operations) for PFD, polynomial factoring, and root calculation. In particular, the time bounds for computing high accuracy PFDs, high accuracy factorizations, and high accuracy approximations for zeros of squarefree polynomials are linear in the output size (and hence optimal) up to logarithmic factors.

The question of finitely many steps in polynomial ideal theory

Grete Hermann's 1926 paper Die Frage der endlich vielen Schritte in der Theorie der Polynomideale, which appears below in English translation, is an intriguing example of ideas before their time. While computational aspects of mathematics were more fashionable before the abstractions of the twentieth century took hold, mathematicians of that time certainly knew nothing of computers nor of today's idea of what an algorithm is. The significance of the paper can be found on the first page, where we find (in translation):The claim that a computation can be found in finitely many steps will mean here that an upper bound for the number of necessary operations for the computation can be specified. Thus it is not enough, for example, to suggest a procedure, for which it can be proved theoretically that it can be executed in finitely many operations, if no upper bound for the number of operations is known.The fact that the author requires an upper bound suggests that there must exist an actual procedure or algorithm for doing computations. We see in this paper the first examples of procedures (with upper bounds given) for a variety of computations in multivariate polynomial ideals. Thus we have here a paper anticipating by 39 years the birth of computer algebra (generally marked by Buchberger's invention of Gröbner bases in his 1965 Ph.D. thesis). The computational procedures which are presented in this paper include multivariate polynomial factorization, polynomial system solving, least common multiples, greatest common divisors, ideal quotients, divisibility of one ideal by another, fundamental ideals, norms, elementary divisor forms, associated prime ideals, primary decomposition, and isolated components. The statements of the theorems frequently end with "can be computed in finitely many steps". The proofs then outline the methods and compute the bounds. One would not at all expect a 1926 paper to contain optimal algorithms, but the fact that procedures for these computations even existed in 1926 make this paper worthy to be remembered.

Fluctuation bounds for sock-sorting and other stochastic processes

Using standard methods from empirical-process theory, in particular symmetrization, we derive exponential bounds on the fluctuations of stochastic processes which may be represented as the averages of many small functions. As examples, self-service queueing and storage problems are analyzed. We eliminate some of the very large constants or polynomial factors which have appeared in other, more asymptotically oriented results. The range of applications is extended by our replacing the more usual covering-number bounds by a dependence on the total variation of the component functions. While this restricts our results to the one-dimensional context, it allows the bounds to be applied to cases in which the component functions are not jump-functions and not of a uniform shape. We also dispense with the assumption that the functions are identically distributed.

A Poisson * Negative Binomial Convolution Law for Random Polynomials over Finite Fields

Let Fq[X] denote a polynomial ring over a finite field Fq with q elements. Let 𝒫n be the set of monic polynomials over Fq of degree n. Assuming that each of the qn possible monic polynomials in 𝒫n is equally likely, we give a complete characterization of the limiting behavior of P(Ωn=m) as n→∞ by a uniform asymptotic formula valid for m≥1 and n−m→∞, where Ωn represents the number (multiplicities counted) of irreducible factors in the factorization of a random polynomial in 𝒫n. The distribution of Ωn is essentially the convolution of a Poisson distribution with mean log n and a negative binomial distribution with parameters q and q−1. Such a convolution law exhibits three modes of asymptotic behaviors: when m is small, it behaves like a Poisson distribution; when m becomes large, its behavior is dominated by a negative binomial distribution, the transitional behavior being essentially a parabolic cylinder function (or some linear combinations of the standard normal law and its iterated integrals). As applications of this uniform asymptotic formula, we derive most known results concerning P(Ωn=m) and present many new ones like the unimodality of the distribution. The methods used are widely applicable to other problems on multiset constructions. An extension to Rényi's problem, concerning the distribution of the difference of the (total) number of irreducibles and the number of distinct irreducibles, is also presented. © 1998 John Wiley & Sons, Inc. Random Struct. Alg., 13, 17–47, 1998

A Poisson * Negative Binomial Convolution Law for Random Polynomials over Finite Fields

Let Fq[X] denote a polynomial ring over a finite field Fq with q elements. Let n be the set of monic polynomials over Fq of degree n. Assuming that each of the qn possible monic polynomials in n is equally likely, we give a complete characterization of the limiting behavior of P(Ωn=m) as n∞ by a uniform asymptotic formula valid for m≥1 and n−m∞, where Ωn represents the number (multiplicities counted) of irreducible factors in the factorization of a random polynomial in n. The distribution of Ωn is essentially the convolution of a Poisson distribution with mean log n and a negative binomial distribution with parameters q and q−1. Such a convolution law exhibits three modes of asymptotic behaviors: when m is small, it behaves like a Poisson distribution; when m becomes large, its behavior is dominated by a negative binomial distribution, the transitional behavior being essentially a parabolic cylinder function (or some linear combinations of the standard normal law and its iterated integrals). As applications of this uniform asymptotic formula, we derive most known results concerning P(Ωn=m) and present many new ones like the unimodality of the distribution. The methods used are widely applicable to other problems on multiset constructions. An extension to Renyi's problem, concerning the distribution of the difference of the (total) number of irreducibles and the number of distinct irreducibles, is also presented. © 1998 John Wiley & Sons, Inc. Random Struct. Alg., 13, 17–47, 1998

Multidimensional Spectral Factorization Through the Root Perturbation Method of Multidimensional Polynomial Factorization

The Spectral Factorization problem (in m Dimensions) has been solved recently by the author via the reduction method of multidimensional (m-D) polynomial factorization. Here, the same problem is also solved through the root perturbation method which is a recently proposed m-D polynomial factorization method.

Numerical Computation of Cross-Covariance Functionals for Linear Systems With Multiple Time Delays

The problem of evaluating a cross-covariance functional of the form 12πi∫-i∞i∞F(s)G(-s)ds arises, e.g., in parametric optimization oflinear systems and in optimal linear filtering. In this paper, an algorithm is presented for computing cross-covariance functionals for linear systems with multiple commensurate time delays. The algorithm is based on converting a crosscovariance functional into an equivalent auto-covariance one. Then by utilizing an existing partialfraction-expansion scheme for a class of two-dimensional rational transfer functions, the evaluation ofa cross-covariance functional is reduced to the determination ofsymbolic inversion ofa polynomial matrix and the spectral factorization of a finite-degree real-coefficient polynomial. It is shown that the presented algorithm can be extended to systems having non-commensurate time delays through either introducing fictitious delays or adjusting time delays with rational approximation such that the resulted delays are commensurate. In order to illustrate the effectiveness of the algorithm, an example is provided.

A Geometric Condition for a Hyperplane Arrangement to be Free

LetG(r, 1, l) be the complex arrangement {xi, xj−ξhxk}, whereξis a primitiverth root of unity. The matroids of these arrangements are the Dowling matroidsQl(Zr), whereZris the group ofrth roots of unity. We show that ifEis a subset ofG(r, 1, l) which does not contain any matroid line, then deletingEgives an arrangementG(r, 1, l)\\Ewhich is free. We also give necessary and sufficient conditions on a set E containing at least one matroid line but no matroid planes so that the deletionG(r, 1, l)\\Ehas a characteristic polynomial which factors completely over the integers. Two types of arrangements can be obtained in this way. We show that one type is always non-free. This yields examples of non-free complex arrangements whose characteristic polynomials factor completely over the integers. The same ideas also yield examples of non-free arrangements over any sufficiently large field (and hence, over the reals) with characteristic polynomials which factor completely over the integers. The matroids of these arrangements are non-supersolvable matroids whose characteristic polynomials factor completely over the integers.