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

Let D = pq be the product of two distinct odd primes. Assuming the parity conjecture, we construct infinitely many r ⩾ 1 such that E
 2rD
 : y
 2 = x
 3 −2rDx has conjectural rank one and v
 p
 (x([k]Q)) ≠ v
 q
 (x([k]Q)) for any odd integer k, where Q is the generator of the free part of E(ℚ). Furthermore, under the generalized Riemann hypothesis, the minimal value of r is less than c log4
 D for some absolute constant c. As a corollary, one can factor D by computing the generator Q.

On the elliptic curve y 2 = x 3 −2 rDx and factoring integers

This dissertation revolves around the BSD (Birch and Swinnerton-Dyer) conjecture for elliptic curves defined over the rational numbers, a famous problem that has been open for over forty years and one of the seven Millennium Prize problems. The BSD conjecture is considered to be the first nontrivial number theoretic problem put forth as a result of explicit machine computation — in the late ’50s at Cambridge University. The BSD conjecture relates the rank of the Mordell-Weil group, the group of rational points of an elliptic curve, a quantity which seems to be difficult to pin down, to the order of vanishing of the L-function of the elliptic curve at its central point. We make algorithmic and theoretical advances with regards to some of the terms appearing in the BSD formula, namely the sizes of the torsion subgroup and the ShafarevichTate group. Firstly, we introduce an algorithm to compute elliptic curve torsion subgroup. The randomized version of this procedure runs in expected time which is essentially linear in the number of bits required to write down the equation of the elliptic curve. Next, we discuss a conjecture of Hindry, who proposed a Brauer-Siegel type formula for elliptic curves. Driven by a suggestion of Hindry, we prove assuming standard conjectures that there are infinitely many elliptic curves with Shafarevich-Tate group of size about

Some computational problems motivated by the Birch and Swinnerton -Dyer conjecture

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Some computational problems motivated by the Birch and Swinnerton -Dyer conjecture