Algebraic Geometry and its Applications

Algebraic Geometry and its Applications will be of interest not only to mathematicians but also to computer scientists working on visualization and related topics. The book is based on 32 invited papers presented at a conference in honor of Shreeram Abhyankar's 60th birthday, which was held in June 1990 at Purdue University and attended by many renowned mathematicians (field medalists), computer scientists and engineers. The keynote paper is by G. Birkhoff; other contributors include such leading names in algebraic geometry as R. Hartshorne, J. Heintz, J.I. Igusa, D. Lazard, D. Mumford, and J.-P. Serre.

I Past, Present, and Future.- Mathematics and India.- 1.1 Mathematics in Europe.- 1.2 Mathematics in India up to Ramanujan.- 1.3 Indology and Some Post-Ramanujan Development.- 1.4 Conclusion.- 1.5 References.- II Algebraic Curves.- 2 Square-root Parametrization of Plane Curves.- 2.1 Introduction.- 2.2 Hyperelliptic Curves.- 2.3 Deriving the Special Polynomial.- 2.4 Singularities of the Auxiliary Curve.- 2.5 Parametrizing the Auxiliary Curve in Characteristic Seven..- 2.6 Finding the factors.- 2.7 The factorization.- 2.8 Galois groups.- 2.9 References.- 3 A Letter as an Appendix to the Square-Root Parameterization Paper Of Abhyankar.- The Letter.- 4 Equisingularity Invariants of Plane Curves.- 4.1 Introduction.- 4.2 Equiresolution Class.- 4.3 Apery Basis Relative to a Parameter.- 4.4 Technical Lemmas.- 4.5 Inversion.- 4.6 Apery Basis and Formal Quadratic Transformations..- 4.7 Algebroid Case.- 4.8 References.- 5 Classification of Algebraic Space Curves, III.- 5.1 Introduction.- 5.2 The Classification.- 5.3 Examples.- 5.4 Open Problems.- 5.5 References.- 6 Plane Polynomial Curves.- 6.1 Introduction.- 6.2 Preliminaries and Notation.- 6.3 Semigroups of Curves with One Place at Infinity.- 6.4 Degree Semigroups of Polynomial Curves.- 6.5 References.- III Algebraic Surfaces.- 7 A Sharp Castelnuovo Bound for the Normalization of Certain Projective Surfaces.- 7.1 Introduction.- 7.2 References.- 8 Abhyankar’s Work on Desingularization.- 8.1 Introduction.- 8.2 Algebraic Curves.- 8.3 Algebraic Surfaces.- 8.4 References.- 9 Moduli Spaces for Special Surfaces of General Type.- 9.1 Introduction.- 9.2 Deformations of Surfaces.- 9.3 Moduli Space of Surfaces.- 9.4 References.- IV Analytic Functions.- 10 A Stationary Phase Formula for p-ADIC Integrals and its Applications.- 10.1 Introduction.- 10.2 p-adic Stationary Phase Formula.- 10.3 Orbital Structure over Fq.- 10.4 Ten Partial Integrals and Z(s).- 10.5 Computation of I1, I1?, I2, I2?, I2?, I4?.- 10.6 Computation of I3.- 10.7 Computation of I4?.- 10.8 Computation of I4.- 10.9 Computation of I5.- 10.10 References.- V Groups and Coverings.- 11 The Q-admissibility of 2A6 and 2A7.- 11.1 Introduction.- 11.2 Notation.- 11.3 Some Polynomials.- 11.4 The Proof of Theorem A.- 11.5 References.- 12 Groups Which Cannot be Realized as Fundamental Groups of the Complements to Hypersurfaces in CN.- 12.1 Introduction.- 12.2 Alexander Polynomials of Plane Curves.- 12.3 References.- 13 Unramified Coverings of the Affine Line in Positive Characteristic.- 13.1 Introduction.- 13.2 Unramified Coverings.- 13.3 A remark on the Lang morphism.- VI Young Tableaux.- 14 Abhyankar’s Work on Young Tableaux and Some Recent Developments.- 14.1 Introduction.- 14.2 Preliminaries plus Preview.- 14.3 Enumeration of Standard Tableaux.- 14.4 Universal Determinantal Identity.- 14.5 Indexed Monomials.- 14.6 Determinantal Ideals and their Hilbert Functions.- 14.7 Some Recent Developements.- 14.8 References.- Abhyankar’s Recursive Formula Regarding Standard Bi-Tableau.- 15.1 Notation and definitions.- 15.2 Integer Valued Functions FD(LK)(m,p,a).- 15.3 Abhyankar’s Recursive formula.- 15.4 Some Results.- 15.5 References.- 16 Correspondences Between Tableaux and Monomials.- 16.1 Introduction.- 16.2 Notation and Terminology.- 16.3 Generalized Rodeletive Correspondence.- 16.4 Generalized Codeletion.- 16.5 Generalized Roinsertion.- 16.6 Generalized Coinsertion.- 16.7 Applications.- 16.8 References.- VII Commutative Algebra.- 17 Report on the Torsion of the Differential Module of an Algebraic Curve.- 17.1 Introduction.- 17.2 Conditions on the Number of Generators of I.- 17.3 Exact Differentials, Maximal Torsion and Quasi Homogeneous Singularities.- 17.4 Conditions on the Embedding Dimension, The Index of Stability,and the Multiplicity.- 17.5 Conditions on the Linkage Class.- 17.6 Smoothability Conditions.- 17.7 Quadratic Transforms.- 17.8 Equisingularity.- 17.9 References.- 18 A Quick Proof of the Hartshorne―Lichtenbaum Vanishing Theorem.- 18.1 Introduction.- 18.2 The Proof.- 18.3 References.- 19 Projective Lines Over One-Dimensional Semilocal Domains and Spectra of Birational Extensions.- 19.1 Introduction.- 19.2 The Projective Line Over a One-Dimensional Semilocal Domain.- 19.3 Spectra of Birational Extensions of the Affine Line.- 19.4 Spectra of Parameter Blowups of Two-Dimensional Local Domains.- 19.5 References.- 20 Some Questions on Z[?14].- 20.1 Introduction.- 20.2 Notation and Terminology.- 20.3 Some questions.- 20.4 Observation.- 20.5 Bases of the conjecture.- 20.6 Remark.- 20.7 References.- 21 Function Fields of Conies, a Theorem of Amitsur―MacRae, and a Problem of Zariski.- 21.1 Introduction.- 21.2 Function fields.- 21.3 The canonical defining conics.- 21.4 Splitting.- 21.5 The Amitsur―MacRae theorem.- 21.6 The Zariski problem.- 21.7 Bibliographic remarks and References.- 21.8 References.- 22 Gradings of Polynomial Rings.- 22.1 Introduction.- 22.2 The Question.- 22.3 Homogeneous Maximal Ideals.- 22.4 Maximal Homogeneous Ideals.- 22.5 Some Answers.- 22.6 References.- 23 Rigid Hilbert Polynomials for ra-Primary Ideals.- 23.1 Introduction.- 23.2 Rigidity of Polynomials.- 23.3 References.- 24 One-Dimensional Local Rings with Finite Cohen―Macaulay Type.- 24.1 Introduction.- 24.2 Necessary and Sufficient Conditions.- 24.3 Degree 3 Extensions.- 24.4 References.- VIII Computational Algebraic Geometry.- 25 Some Applications of Constructive Real Algebraic Geometry.- 25.1 Introduction.- 25.2 Global Parameterization.- 25.3 Local Parameterization.- 25.4 Intersection.- 25.5 Interpolation and Approximation.- 25.6 References.- 26 An Improved Sign Determination Algorithm.- 26.1 Introduction.- 26.2 Sign Determination.- 26.3 A Sign-Determination Algorithm.- 26.4 Conclusions.- 26.5 References.- 27 Decomposition Algorithms in Geometry.- 27.1 Introduction.- 27.2 The Two-Dimensional Case.- 27.3 The Three-dimensional Case.- 27.4 Concluding Remarks.- 27.5 References.- 28 Single Exponential Path Finding in Semi-algebraic Sets, Part II: The General Case.- 28.1 Introduction.- 28.2 Some auxiliary results.- 28.3 Proof of the Main Theorem.- 28.4 A note about the computation of the connected components.- 28.5 References.- 29 An Improved Projection for Cylindrical Algebraic Decomposition.- 29.1 Introduction.- 29.2 Working domains and basic definitions.- 29.3 The projection set.- 29.4 The CAD algorithm.- 29.5 Practical improvements.- 29.6 References.- 30 Degree Bounds of Gröbner Bases.- 30.1 Introduction.- 30.2 Preliminaries.- 30.3 Nonexistence of Bounds Over Z[x, y].- 30.4 Some Problems on Complexity.- 30.5 References.- 31 Elastica and Computer Vision.- 31.1 Introduction.- 31.2 Edges in Computer Vision.- 31.3 A Brownian Prior for Edges.- 31.4 Alternate Priors.- 31.5 The Differential Equation of Elastica.- 31.6 Solving for Elastica.- 31.7 References.- 32 Isolator Polynomials.- 32.1 Introduction.- 32.2 Isolator Polynomials.- 32.3 Motivating Examples.- 32.4 Polynomial Remainder Sequences.- 32.5 Conclusion.- 32.6 References.- 33 A Bound on the Implicit Degree of Polygonal Bézier Surfaces.- 33.1 Introduction.- 33.2 Base points.- 33.3 Intersection multiplicity and Newton polygons.- 33.4 A degree bound for polygonal patches.- 33.5 References.- IX Publications of Shreeram.