# Algebraic Geometry for Scientists and Engineers

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*Shreeram S. Abhyankar*

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

# Table of Contents

## Algebraic Geometry for Scientists and Engineers

- Contents ix10 free
- Preface xi12 free
- Lecture 1. Rational and Polynomial Parametrizations 116 free
- Lecture 2. Fractional Linear Transformations 1126
- Lecture 3. Cubic Curves 1732
- Lecture 4. Cubic Surfaces and General Hypersurfaces 2338
- Lecture 5. Outline of the Theory of Plane Curves 3348
- Lecture 6. Affine Plane and Projective Plane 4156
- Lecture 7. Sphere with Handles 4762
- Lecture 8. Functions and Differentials on a Curve 5772
- Lecture 9. Polynomials and Power Series 6782
- Lecture 10. Review of Abstract Algebra 7590
- Lecture 11. Some Commutative Algebra 8398
- Lecture 12. Hensel's Lemma and Newton's Theorem 89104
- Lecture 13. More about Newton's Theorem 95110
- Lecture 14. Branches and Valuations 99114
- Lecture 15. Divisors of Functions and Differentials 109124
- Lecture 16. Weierstrass Preparation Theorem 119134
- Lecture 17. Intersection Multiplicity 125140
- Lecture 18. Resolution of Singularities of Plane Curves 131146
- Lecture 19. Infinitely Near Singularities 145160
- Lecture 20. Parametrizing a Quartic with Three Double Points 159174
- Lecture 21. Characteristic Pairs 165180
- Lecture 22. Criterion For One Place and Jacobian Problem 177192
- Lecture 23. Inversion Formula and Jacobian Problem 189204
- Lecture 24. Surfaces 195210
- Lecture 25. Hypersurfaces 209224
- Lecture 26. Resolution of Singularities of Algebraic Surfaces 221236
- Lecture 27. Birational and Polyrational Transformations 235250
- Lecture 28. Valuations and Birational Correspondence 243258
- Lecture 29. Rational Cylinders through a Variety 255270
- Lecture 30. Resultants 267282
- Bibliography 275290
- Index 283298