Algebraic Geometry and Its Applications

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.