Mathematics for Computer Algebra

Mathematics for Computer Algebra

Paperback(Softcover reprint of the original 1st ed. 1992)

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Product Details

ISBN-13: 9781461391739
Publisher: Springer New York
Publication date: 10/14/2011
Edition description: Softcover reprint of the original 1st ed. 1992
Pages: 346
Product dimensions: 6.10(w) x 9.25(h) x 0.03(d)

Table of Contents

1 Elementary Arithmetics.- 1. Representation of an integer in basis B1.- 1. Lexicographical order.- 2. Development in basis B, existence.- 3. Prom development to number.- 4. Prom number to development in basis B.- 5. Case of general rational numbers.- 6. Comparing two numbers.- 2. Addition.- 1. Case of two positive numbers.- 2. Case of two numbers of any sign.- 3. Subtraction.- 4. Multiplication.- 5. Euclidean division.- 1. Existence.- 2. Computation.- 6. The cost of multiplication and division.- 1. The cost of multiplication.- 2. The cost of division.- 7. How to compute powers.- 1. First algorithm.- 2. Second algorithm.- 3. One application.- 4. Complexity of this problem.- 8. The g.c.d..- 1. Existence. Relation of Bézout. Theorem of Euclid-Gauss.- 2. How to compute the g.c.d..- 3. The cost of the algorithm of Euclid.- 4. The l.c.m..- 9. The group G (n).- 1. Definition.- 2. The theorem of Euler.- 3. Computation of the inverse in G (n).- 4. Computation of the coefficients of the relation of Bézout.- 10. The Chinese remainder theorem.- 1. The theorem.- 2. Constructive proof.- 3. Effective computations.- 11. The prime numbers.- 2 Number Theory, Complements.- 1. Study of the group G(n).- 1. Some lemmas on finite groups.- 2. Application to G(p).- 3. Structure of G(pk), k— 2, p odd prime.- 4. Structure of G (2k).- 5. Structure of G (n).- 2. Tests of primality.- 1. A general theorem.- 2. A simple test of primality.- 3. Elementary tests.- 4. Statistics on G (n).- 3. Factorization of rational integers.- 1. Methods by successive divisions.- 2. Method of Fermat.- 3. Method of Sherman Lehman.- 4. Pollard’s rho method.- 3 Polynomials, Algebraic Study.- 1. Definitions and elementary properties.- 1. First definitions.- 2. Elementary arithmetic operations.- 3. Notions of degree.- 4. Case of an integral domain.- 2. Euclidean division.- 1. Monic polynomials.- 2. Euclidean division.- 3. The case of a field.- 4. Pseudo-division.- 3. The Chinese remainder theorem.- 4. Factorization.- 1. Case of a field.- 2. Case of a factorial ring.- 5. Polynomial functions.- 1. Definition.- 2. Roots of a polynomial.- 3. Multiplicity of a root.- 4. Derivatives and roots.- 5. Taylor’s formula.- 6. The resultant.- 7. Companion matrix.- 8. Linear recursive sequences.- 4 Polynomials with complex coefficients.- 1. The theorem of d’Alembert.- 1. Statement.- 2. Analytic properties.- 3. Demonstration of the theorem.- 4. Irreducible real polynomials.- 2. Estimates of the roots.- 1. Principle of the demonstration.- 2. An analytic lemma.- 3. Bounds for the roots.- 3. The measure of a polynomial.- 1. Definition.- 2. An algebraic lemma.- 3. An upper bound for M(P).- 4. Other upper bounds.- 5. Analytic results.- 6. A method for the computation of M(P).- 7. An example of the evaluation of M (P).- 4. Bounds for size of the factors of a polynomial.- 1. Definitions.- 2. Upper bounds for the factors in the case of a single variable.- 3. Definition of the measure in the case of several variables.- 4. Bounds for the factors of a polynomial, case of several variables.- 5. An example in the case of one variable.- 5. The distribution of the roots of a polynomial.- 1. An upper bound for the number of real roots.- 2. Distribution of the arguments of the roots of a polynomial.- 6. Separation of the roots of a polynomial.- 1. Notations.- 2. A lower bound for sep(P).- 3. Other lower bounds for the distance between two roots.- 4. Use of Galois properties.- 5. An example.- 5 Polynomials with real coefficients.- 1. Polynomials irreducible over—.- 2. The theorem of Rolle.- 1. The theorem of intermediate values.- 2. The theorem of Rolle.- 3. Estimates of real roots.- 1. The rule of Newton.- 2. The rule of Lagrange and MacLaurin.- 3. A special case of the rule of Descartes.- 4. The rule of Cauchy.- 5. An example of an estimation of the real roots of a polynomial.- 4. The number of zeros of a polynomial in a real interval.- 1. The rule of Sturm.- 2. The theorem of Budan-Fourier.- 3. The rule of Descartes.- 4. Case of the polynomials whose all roots are real.- 5. A detailed example.- 6. Vincent’s theorem.- 5. Equations whose roots have a negative real part.- 6/Polynomials over finite fields.- 1. Finite fields.- 1. General results.- 2. The operations in a finite field.- 3. Determination of a primitive element of H*.- 4. Determination of z such that Wq—Hp[z].- 5. An example: H8.- 2. Statistics on Hq[X].- 1. The function of Möbius.- 2. Counting irreducible polynomials.- 3. Number of squarefree polynomials.- 4. Study of the number of irreducible factors of a polynomial.- 3. Factorization into a product of squarefree polynomials.- 1. Definitions and generalites.- 2. Case of characteristic zero.- 3. Case of non zero characteristic.- 4. Algorithms of decomposition into a product of squarefree polynomials.- 4. Factorization of the polynomials over a finite field.- 1. Determination of the number of irreducible factors.- 2. Complete factorization.- 3. Cost of the algorithm.- 4. Case where q is large.- 5. Decomposition into a product of irreducible factors of given degree.- 6. The method of MacEleice.- 5. Search for the roots of a polynomial in a finite field.- 1. A first reduction.- 2. Case of the polynomials which split completely in Hq.- 3. Solution of the binomial equation Xd= a.- 7 Polynomials with integer coefficients.- 1. Principles of the algorithms of factorization.- 1. Existence of an algorithm of factorization.- 2. The algorithm of Kronecker.- 3. The principle of modern algorithms.- 2. The choice of the prime modulus.- 3. Refining the factorization.- 1. The bound B.- 2. Hensel’s lemma.- 4. Berlekamp’s method of factorization.- 5. The algorithm L3.- 1. Lattices.- 2. Orthogonalization of Gram-Schmidt.- 3. Reduced basis.- 4. Algorithm of reduction.- 5. Cost of the algorithm.- 6. Factors of polynomials and lattices.- 7. The algorithm of factorization.- Index of Names.

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