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 oneor 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.

Modern computer algebra

WHAT is the theory of functions about? This question may be heard now and again from a mathematical student; and if, by way of a pattial reply, it be said that the elements of the theory of functions forms the basis on which the whole of that part of pure mathematics which deals with continuously varying quantity rests, the answer would not be too wide nor would it always imply too much. Theory of Functions of a Complex Variable. By Dr. A. R. Forsyth. (Cambridge University Press, 1893.)

Theory of Functions of a Complex Variable

When writing can change your life, when writing can enrich you by offering much money, why don't you try it? Are you still very confused of where getting the ideas? Do you still have no idea with what you are going to write? Now, you will need reading. A good writer is a good reader at once. You can define how you write depending on what books to read. This solved and unsolved problems in number theory can help you to solve the problem. It can be one of the right sources to develop your writing skill.

Solved and Unsolved Problems in Number Theory.

This chapter discusses finite fields and their applications. It also traces the historical development of finite fields, outlining some of their basic properties. In general, F q denotes the finite field of order q . The general theory of finite fields may be said to begin with the work of Carl Friedrich Gauss and Evariste Galois. Galois supposes Fx to be irreducible mod p and of degree v and solves Fx ≡ 0 by introducing new symbols, which might be just as useful as the imaginary unit i in analysis. Galois approach via imaginary roots and Dedekind's approach via residue class rings are shown to be essentially equivalent by Kronecker. A careful choice of the representation of a finite field F q may assist in the algorithms for the implementation of arithmetic operations in F q . Enumeration theorems for ordered bases of various types are known. Irreducible polynomials of degree n over F q are important for the construction of the field F q n .

/pdf/finite-fields-and-their-applications-26qwtmy463.pdf

Finite fields and their applications

Two high order multi-augmented and improved augmented finite volume methods are proposed for solving nonlinear degenerate parabolic problems. The solution is represented as Puiseux series expansion in a subdomain with singularity, but contains undetermined parameters called augmented variables. The equation in regular subdomain is treated with high accuracy numerical methods and the unknown parameters can be solved simultaneously from the corresponding nonlinear system. The outstanding advantages of the proposed methods are that the degeneracy can be depicted by the semi-analytic solution, and we can get high order results globally. Specially, the convergence order for nonlinear degenerate parabolic problems is determined by the numerical schemes on regular subdomain, and the augmented methods have good robustness for solving degenerate or singular problems. Numerical examples for some degenerate parabolic equations confirm the efficiency of the new methods including the second and fourth order schemes. In particular, a two-dimensional singular elliptic equation with corner degeneracy is presented in the numerical experiments to demonstrate that the proposed methods can be extended to higher dimensional degenerate problems.

The High Order Augmented Finite Volume Methods Based on Series Expansion for Nonlinear Degenerate Parabolic Equations

https://hal.archives-ouvertes.fr/hal-00825850/document

Good reduction of Puiseux series and applications

Let K be a field of characteristic p with q elements and FE in L[X,Y] be a polynomial with p> deg_Y(F) and total degree d. In [40], we showed that rational Puiseux series of F above X=0 could be computed with an expected number of O~(d5+d3log q) arithmetic operations in L. In this paper, we reduce this bound to O~(d4+d2log q) using Hensel lifting and changes of variables in the Newton-Puiseux algorithm that give a better control of the number of steps. The only asymptotically fast algorithm required is polynomial multiplication over finite fields. This approach also allows to test the irreducibility of F in L[[X]][Y] with O(d3) operations in K. Finally, we describe a method based on structured bivariate multiplication [34] that may speed up computations for some input.

/pdf/improving-complexity-bounds-for-the-computation-of-puiseux-443yn7zv9w.pdf

Improving Complexity Bounds for the Computation of Puiseux Series over Finite Fields

In [12], we sketched a numeric-symbolic method to compute Puiseux series with floating point coefficients. In this paper, we address the symbolic part of our algorithm. We study the reduction of Puiseux series coefficients modulo a prime ideal and prove a good reduction criterion sufficient to preserve the required information, namely Newton polygon trees. We introduce a convenient modification of Newton polygons that greatly simplifies proofs and statements of our results. Finally, we improve complexity bounds for Puiseux series calculations over finite fields, and estimate the bit-complexity of polygon tree computation.

Good reduction of puiseux series and complexity of the Newton-Puiseux algorithm over finite fields

Search of primitive polynomials over finite fields

We have designed a new symbolic-numeric strategy to compute efficiently and accurately floating point Puiseux series defined by a bivariate polynomial over an algebraic number field. In essence, computations modulo a well chosen prime $p$ are used to obtain the exact information required to guide floating point computations. In this paper, we detail the symbolic part of our algorithm: First of all, we study modular reduction of Puiseux series and give a good reduction criterion to ensure that the information required by the numerical part is preserved. To establish our results, we introduce a simple modification of classical Newton polygons, that we call "generic Newton polygons", which happen to be very convenient. Then, we estimate the arithmetic complexity of computing Puiseux series over finite fields and improve known bounds. Finally, we give bit-complexity bounds for deterministic and randomized versions of the symbolic part. The details of the numerical part will be described in a forthcoming paper.

Towards a Symbolic-Numeric Method to Compute Puiseux Series: The Modular Part

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