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

First published in 1975, this classic book gives a systematic account of transcendental number theory, that is those numbers which cannot be expressed as the roots of algebraic equations having rational coefficients. Their study has developed into a fertile and extensive theory enriching many branches of pure mathematics. Expositions are presented of theories relating to linear forms in the logarithms of algebraic numbers, of Schmidt's generalisation of the Thue-Siegel-Roth theorem, of Shidlovsky's work on Siegel's |E|-functions and of Sprindzuk's solution to the Mahler conjecture. The volume was revised in 1979: however Professor Baker has taken this further opportunity to update the book including new advances in the theory and many new references.

Transcendental number theory, by Alan Baker. Pp. x, 147. £4·90. 1975. SBN 0 521 20461 5 (Cambridge University Press)

Theory of Equations. By J. V. Uspensky. Pp. vii, 353. 27s. 1948. (McGraw-Hill)

Proceedings of the American Mathematical Society

Functions of one complex variable II

We give a unified ("basis free") framework for the Descartes method for real root isolation of square-free real polynomials. This framework encompasses the usual Descartes' rule of sign method for polynomials in the power basis as well as its analog in the Bernstein basis. We then give a new bound on the size of the recursion tree in the Descartes method for polynomials with real coefficients. Applied to polynomials A(X) = Eni=0aiXi with integer coefficients |ai|

https://domino.mpi-inf.mpg.de/intranet/ag1/ag1publ.nsf/d8ab60b7b27d8142c12569e30040d645/6e09cb7fbe62ea64c12571b20040a5d4/$file/esy-desctree-issac06-authprep.pdf

Almost tight recursion tree bounds for the Descartes method

Nonrobustness refers to qualitative or catastrophic failures in geometric algorithms arising from numerical errors. Section 45.1 provides background on these problems. Although nonrobustness is already an issue in “purely numerical” computation, the problem is compounded in “geometric computation.” In Section 45.2 we characterize such computations. Researchers trying to create robust geometric software have tried two approaches: making fixed-precision computation robust (Section 45.3), and making the exact approach viable (Section 45.4). Another source of nonrobustness is the phenomenon of degenerate inputs. General methods for treating degenerate inputs are described in Section 45.5. For some problems the exact approach may be expensive or infeasible. To ensure robustness in this setting, a recent extension of exact computation, the so-called “soft exact approach,” has been proposed. This is described in Section 45.6.

Robust Geometric Computation.

This paper presents two results on the complexity of root isolation via Sturm sequences. Both results exploit amortization arguments.

Amortized Bound for Root Isolation via Sturm Sequences

Complexity of real root isolation using continued fractions

Let F(z) be an arbitrary complex polynomial. We introduce the {local root clustering problem}, to compute a set of natural epsilon-clusters of roots of F(z) in some box region B0 in the complex plane. This may be viewed as an extension of the classical root isolation problem. Our contribution is two-fold: we provide an efficient certified subdivision algorithm for this problem, and we provide a bit-complexity analysis based on the local geometry of the root clusters.Our computational model assumes that arbitrarily good approximations of the coefficients of F(z) are provided by means of an oracle at the cost of reading the coefficients. Our algorithmic techniques come from a companion paper [3] and are based on the Pellet test, Graeffe and Newton iterations, and are independent of Schonhage's splitting circle method. Our algorithm is relatively simple and promises to be efficient in practice.

https://dl.acm.org/doi/pdf/10.1145/2930889.2930939

Vikram Sharma

Papers

Almost tight recursion tree bounds for the Descartes method

Robust Geometric Computation.

Amortized Bound for Root Isolation via Sturm Sequences

Complexity of real root isolation using continued fractions

Complexity Analysis of Root Clustering for a Complex Polynomial