Mathematical Surveys and Monographs 1990; 295 pp; softcover Volume: 35 Reprint/Revision History: fourth printing 2000 ISBN10: 0821815350 ISBN13: 9780821815359 List Price: US$45 Member Price: US$36 Order Code: SURV/35
 This book, based on lectures presented in courses on algebraic geometry taught by the author at Purdue University, is intended for engineers and scientists (especially computer scientists), as well as graduate students and advanced undergraduates in mathematics. In addition to providing a concrete or algorithmic approach to algebraic geometry, the author also attempts to motivate and explain its link to more modern algebraic geometry based on abstract algebra. The book covers various topics in the theory of algebraic curves and surfaces, such as rational and polynomial parametrization, functions and differentials on a curve, branches and valuations, and resolution of singularities. The emphasis is on presenting heuristic ideas and suggestive arguments rather than formal proofs. Readers will gain new insight into the subject of algebraic geometry in a way that should increase appreciation of modern treatments of the subject, as well as enhance its utility in applications in science and industry. Table of Contents  Rational and polynomial parametrizations
 Fractional linear transformations
 Cubic curves
 Cubic surfaces and general hypersurfaces
 Outline of the theory of plane curves
 Affine plane and projective plane
 Sphere with handles
 Functions and differentials on a curve
 Polynomials and power series
 Review of abstract algebra
 Some commutative algebra
 Hensel's lemma and Newton's theorem
 More about Newton's theorem
 Branches and valuations
 Divisors of functions and differentials
 Weierstrass preparation theorem
 Intersection multiplicity
 Resolution of singularities of plane curves
 Infinitely near singularities
 Parametrizing a quartic with three double points
 Characteristic pairs
 Criterion for one place and Jacobian problem
 Inversion formula and Jacobian problem
 Surfaces
 Hypersurfaces
 Resolution of singularities of algebraic surfaces
 Birational and polyrational transformations
 Valuations and birational correspondence
 Rational cylinders through a variety
 Resultants
