Download or read book On the Complexity of Polynomial Factorization Over P adic Fields written by Olga Erzsébet Veres and published by . This book was released on 2009 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Let p be a rational prime and?(x) be a monic irreducible polynomial in Z p [x]. Based on the work of Ore on Newton polygons (Ore, 1928) and MacLane's characterization of polynomial valuations (MacLane, 1936), Montes described an algorithm for the decomposition of the ideal [Special characters omitted.] over an algebraic number field (Montes, 1999). We give a simplified version of the Montes algorithm with a full MAPLE implementation which tests the irreducibility of?(x) over Q p . We derive an estimate of the complexity of this simplified algorithm in the worst case, when?(x) is irreducible over Q p . We show that in this case the algorithm terminates in at most[Special characters omitted.] bit operations. Lastly, we compare the "one-element" and "two-element" variations of the Zassenhaus "Round Four" algorithm with the Montes algorithm.

Download or read book Algorithmic Number Theory written by Guillaume Hanrot and published by Springer Science & Business Media. This book was released on 2010-07-07 with total page 407 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the refereed proceedings of the 9th International Algorithmic Number Theory Symposium, ANTS 2010, held in Nancy, France, in July 2010. The 25 revised full papers presented together with 5 invited papers were carefully reviewed and selected for inclusion in the book. The papers are devoted to algorithmic aspects of number theory, including elementary number theory, algebraic number theory, analytic number theory, geometry of numbers, algebraic geometry, finite fields, and cryptography.

Download or read book Complexity of Deciding Solvability of Polynomial Equations Over P adic Integers written by Aleksandr L. Chistov and published by . This book was released on 1997 with total page 18 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Effective Polynomial Computation written by Richard Zippel and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 364 pages. Available in PDF, EPUB and Kindle. Book excerpt: Effective Polynomial Computation is an introduction to the algorithms of computer algebra. It discusses the basic algorithms for manipulating polynomials including factoring polynomials. These algorithms are discussed from both a theoretical and practical perspective. Those cases where theoretically optimal algorithms are inappropriate are discussed and the practical alternatives are explained. Effective Polynomial Computation provides much of the mathematical motivation of the algorithms discussed to help the reader appreciate the mathematical mechanisms underlying the algorithms, and so that the algorithms will not appear to be constructed out of whole cloth. Preparatory to the discussion of algorithms for polynomials, the first third of this book discusses related issues in elementary number theory. These results are either used in later algorithms (e.g. the discussion of lattices and Diophantine approximation), or analogs of the number theoretic algorithms are used for polynomial problems (e.g. Euclidean algorithm and p-adic numbers). Among the unique features of Effective Polynomial Computation is the detailed material on greatest common divisor and factoring algorithms for sparse multivariate polynomials. In addition, both deterministic and probabilistic algorithms for irreducibility testing of polynomials are discussed.

Download or read book Finite Fields Theory and Computation written by Igor Shparlinski and published by Springer Science & Business Media. This book was released on 2013-03-09 with total page 532 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is mainly devoted to some computational and algorithmic problems in finite fields such as, for example, polynomial factorization, finding irreducible and primitive polynomials, the distribution of these primitive polynomials and of primitive points on elliptic curves, constructing bases of various types and new applications of finite fields to other areas of mathematics. For completeness we in clude two special chapters on some recent advances and applications of the theory of congruences (optimal coefficients, congruential pseudo-random number gener ators, modular arithmetic, etc.) and computational number theory (primality testing, factoring integers, computation in algebraic number theory, etc.). The problems considered here have many applications in Computer Science, Cod ing Theory, Cryptography, Numerical Methods, and so on. There are a few books devoted to more general questions, but the results contained in this book have not till now been collected under one cover. In the present work the author has attempted to point out new links among different areas of the theory of finite fields. It contains many very important results which previously could be found only in widely scattered and hardly available conference proceedings and journals. In particular, we extensively review results which originally appeared only in Russian, and are not well known to mathematicians outside the former USSR.

Download or read book Algorithmic Number Theory written by Wieb Bosma and published by Springer Science & Business Media. This book was released on 2000-06-21 with total page 610 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the refereed proceedings of the 4th International Algorithmic Number Theory Symposium, ANTS-IV, held in Leiden, The Netherlands, in July 2000. The book presents 36 contributed papers which have gone through a thorough round of reviewing, selection and revision. Also included are 4 invited survey papers. Among the topics addressed are gcd algorithms, primality, factoring, sieve methods, cryptography, linear algebra, lattices, algebraic number fields, class groups and fields, elliptic curves, polynomials, function fields, and power sums.

Download or read book LATIN 92 written by Imre Simon and published by Springer Science & Business Media. This book was released on 1992-03-11 with total page 564 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains the proceedings of LATIN '92, a theoretical computer science symposium (Latin American Theoretical Informatics) held in S o Paulo, Brazil in April 1992. LATIN is intended to be a comprehensive symposium in the theory of computing, but for this first meeting the following areas were chosen for preferential coverage: algorithms and data structures, automata and formal languages, computability and complexity theory, computational geometry, cryptography, parallel and distributed computation, symbolic and algebraic computation, and combinatorial and algebraic aspects of computer science. The volume includesfull versions of the invited papers by 11 distinguished guest lecturers as well as 32 contributed papers selected from 66 submissions from authors with affiliations in 26 countries.

Download or read book Modern Computer Algebra written by Joachim von zur Gathen and published by Cambridge University Press. This book was released on 2013-04-25 with total page 811 pages. Available in PDF, EPUB and Kindle. Book excerpt: Computer algebra systems are now ubiquitous in all areas of science and engineering. This highly successful textbook, widely regarded as the 'bible of computer algebra', gives a thorough introduction to the algorithmic basis of the mathematical engine in computer algebra systems. Designed to accompany one- or two-semester courses for advanced undergraduate or graduate students in computer science or mathematics, its comprehensiveness and reliability has also made it an essential reference for professionals in the area. Special features include: detailed study of algorithms including time analysis; implementation reports on several topics; complete proofs of the mathematical underpinnings; and a wide variety of applications (among others, in chemistry, coding theory, cryptography, computational logic, and the design of calendars and musical scales). A great deal of historical information and illustration enlivens the text. In this third edition, errors have been corrected and much of the Fast Euclidean Algorithm chapter has been renovated.

Download or read book Efficient Algorithms for Computations with Sparse Polynomials written by Seyed Mohammad Mahdi Javadi and published by . This book was released on 2011 with total page 252 pages. Available in PDF, EPUB and Kindle. Book excerpt: The problem of interpolating a sparse polynomial has always been one of the central objects of research in the area of computer algebra. It is the key part of many algorithms such as polynomial GCD computation. We present a probabilistic algorithm to interpolate a sparse multivariate polynomial over a finite field, represented with a black box. Our algorithm modifies the Ben-Or/Tiwari algorithm from 1988 for interpolating polynomials over rings with characteristic zero to positive characteristics by doing additional probes. To interpolate a polynomial in n variables with t non-zero terms, Zippel's algorithm interpolates one variable at a time using O(ndt) probes to the black box where d bounds the degree of the polynomial. Our new algorithm does O(nt) probes. We provide benchmarks comparing our algorithm to Zippel's algorithm and the racing algorithm of Kaltofen/Lee. The benchmarks demonstrate that for sparse polynomials our algorithm often makes fewer probes. A key advantage in our new algorithm is, unlike the other two algorithms, it can be parallelized efficiently. Our main application for an efficient sparse interpolation algorithm is computing GCDs of polynomials. We are especially interested in polynomials over algebraic function fields. The best GCD algorithm available is SparseModGcd, presented by Javadi and Monagan in 2006. We further improve this algorithm in three ways. First we prove that we can eliminate the trial divisions in positive characteristic. Trial divisions are the bottleneck of the algorithm for denser polynomials. Second, we give a new (and correct) solution to the normalization problem. Finally we will present a new in-place library of functions for computing GCDs of univariate polynomials over algebraic number fields. Furthermore we present an efficient algorithm for factoring multivariate polynomials over algebraic fields with multiple field extensions and parameters. Our algorithm uses Hensel lifting and extends the EEZ algorithm of Wang which was designed for factorization over rationals. We also give a multivariate p-adic lifting algorithm which uses sparse interpolation. This enables us to avoid using poor bounds on the size of the integer coefficients in the factorization when using Hensel lifting. We provide timings demonstrating the efficiency of our algorithm.

Download or read book Randomization Relaxation and Complexity in Polynomial Equation Solving written by Leonid Gurvits and published by American Mathematical Soc.. This book was released on 2011 with total page 230 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume corresponds to the Banff International Research Station Workshop on Randomization, Relaxation, and Complexity, held from February 28-March 5, 2010. It contains a sample of advanced algorithmic techniques underpinning the solution of systems of polynomial equations. The papers are written by leading experts in algorithmic algebraic geometry and examine core topics.

Download or read book Algorithmic Number Theory Efficient algorithms written by Eric Bach and published by MIT Press. This book was released on 1996 with total page 536 pages. Available in PDF, EPUB and Kindle. Book excerpt: Volume 1.

Download or read book Solving Polynomial Equations written by Alicia Dickenstein and published by Springer Science & Business Media. This book was released on 2005-04-27 with total page 433 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides a general introduction to modern mathematical aspects in computing with multivariate polynomials and in solving algebraic systems. It presents the state of the art in several symbolic, numeric, and symbolic-numeric techniques, including effective and algorithmic methods in algebraic geometry and computational algebra, complexity issues, and applications ranging from statistics and geometric modelling to robotics and vision. Graduate students, as well as researchers in related areas, will find an excellent introduction to currently interesting topics. These cover Groebner and border bases, multivariate resultants, residues, primary decomposition, multivariate polynomial factorization, homotopy continuation, complexity issues, and their applications.

Download or read book Algorithms and Discrete Applied Mathematics written by Apurva Mudgal and published by Springer Nature. This book was released on 2021-01-28 with total page 519 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the proceedings of the 7th International Conference on Algorithms and Discrete Applied Mathematics, CALDAM 2021, which was held in Rupnagar, India, during February 11-13, 2021. The 39 papers presented in this volume were carefully reviewed and selected from 82 submissions. The papers were organized in topical sections named: approximation algorithms; parameterized algorithms; computational geometry; graph theory; combinatorics and algorithms; graph algorithms; and computational complexity.

Download or read book Handbook of Finite Fields written by Gary L. Mullen and published by CRC Press. This book was released on 2013-06-17 with total page 1048 pages. Available in PDF, EPUB and Kindle. Book excerpt: Poised to become the leading reference in the field, the Handbook of Finite Fields is exclusively devoted to the theory and applications of finite fields. More than 80 international contributors compile state-of-the-art research in this definitive handbook. Edited by two renowned researchers, the book uses a uniform style and format throughout and

Download or read book Mathematical Foundations of Computer Science 2001 written by Jiri Sgall and published by Springer. This book was released on 2003-08-06 with total page 735 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the refereed proceedings of the 26th International Symposium on Mathematical Foundations of Computer Science, MFCS 2001, held in Marianske Lazne, Czech Republic in August 2001. The 51 revised full papers presented together with 10 invited contributions were carefully reviewed and selected from a total of 118 submissions. All current aspects of theoretical computer science are addressed ranging from mathematical logic and programming theory to algorithms, discrete mathematics, and complexity theory. Besides classical issues, modern topics like quantum computing are discussed as well.

Download or read book Multivariate Polynomial Factorization written by David R. Musser and published by . This book was released on 1974 with total page 56 pages. Available in PDF, EPUB and Kindle. Book excerpt: This paper describes algorithms for factoring a polynomial in one or more variables, with integer coefficients, into factors which are irreducible over the integers. These algorithms are based on the use of factorizations over finite fields and 'Hensel's Lemma construction'. 'Abstract algorithm' descriptions are used in the presentation of the underlying algebraic theory. Included is a new generalization of Hensel's p-adic construction which leads to a practical algorithm for factoring multivariate polynomials. The univariate case algorithm is also specified in greater detail than in the previous literature, with attention to a number of improvements which the author has developed based on theoretical computing time analyses and experience with actual implementations. (Author).

Download or read book Analytic Computational Complexity written by J.F. Traub and published by Academic Press. This book was released on 2014-05-10 with total page 250 pages. Available in PDF, EPUB and Kindle. Book excerpt: Analytic Computational Complexity contains the proceedings of the Symposium on Analytic Computational Complexity held by the Computer Science Department, Carnegie-Mellon University, Pittsburgh, Pennsylvania, on April 7-8, 1975. The symposium provided a forum for assessing progress made in analytic computational complexity and covered topics ranging from strict lower and upper bounds on iterative computational complexity to numerical stability of iterations for solution of nonlinear equations and large linear systems. Comprised of 14 chapters, this book begins with an introduction to analytic computational complexity before turning to proof techniques used in analytic complexity. Subsequent chapters focus on the complexity of obtaining starting points for solving operator equations by Newton's method; maximal order of multipoint iterations using n evaluations; the use of integrals in the solution of nonlinear equations in N dimensions; and the complexity of differential equations. Algebraic constructions in an analytic setting are also discussed, along with the computational complexity of approximation operators. This monograph will be of interest to students and practitioners in the fields of applied mathematics and computer science.