Affine Algebraic Curve Research Articles

Geometry: our cultural heritage

Part I. A Cultural Heritage: Chapter 1. Early Beginnings 1.1 Prehistory 1.2 Geometry in the New Stone Age 1.3 Early Mathematics and Ethnomathematics Chapter 2. The Great River Civilizations 2.1 Civilizations long dead -- and yet alive 2.2 Birth of Geometry as we know it 2.3 Geometry in the Land of the Pharaoh 2.4 Babylonian Geometry Chapter 3. Greek and Hellenic Geometry 3.1 Early Greek Geometry. Thales of Miletus 3.2 The story of Pythagoras and the Pythagoreans 3.3 The Geometry of the Pythagoreans 3.4 The Discovery of Irrational Numbers 3.5 Origin of the Classical Problems 3.6 Constructions by Compass and Straightedge 3.7 Squaring the Circle 3.8 Doubling the Cube 3.9 Trisecting any Angle 3.10 Plato and the Platonic Solids 3.11 Archytas and the Doubling of the Cube Chapter 4. Geometry in the Hellenistic Era 4.1 Euclid and Euclids Elements 4.2 The Books of Euclids Elements 4.3 The Roman Empire 4.4 Archimedes 4.5 Erathostenes and the Duplication of the Cube 4.6 Nicomedes and his Conchoid 4.7 Apollonius and the Conic Sections 4.8 Caesar and the End of the Republic in Rome 4.9 The Murder of Hypatia 4.10 The Decline and Fall of the Roman Empire Chapter 5. The Geometry of Yesterday and Today 5.1 The Dark Middle Ages 5.2 Geometry Reawakening: A new Dawn in Europe 5.3 Elementary Geometry and Higher Geometry 5.4 Desargues and the two Pascals 5.5 Descartes and Analytic Geometry 5.6 Geometry in the 18th. Century 5.6.1 Cramers theorem 5.7 Some Features of Modern Geometry Chapter 6. Geometry and the Real World 6.1 Mathematics and Predicting Catastrophes 6.2 Catastrophe Theory 6.3 Geometric Shapes in Nature 6.4 Fractal Structures in Nature Part II. Introduction to Geometry Chapter 7. Axiomatic Geometry 7.1 The Postulates of Euclid and Hilbert's Explanation 7.2 Non-Euclidian Geometry 7.3 Logic and Intuitive Set Theory 7.4 Axioms, Axiomatic Theories and Models 7.5 General Theory of Axiomatic Systems Chapter 8. Axiomatic Projective Geometry 8.1 Plane Projective Geometry 8.2 An Unsolved Geometric Problem 8.3 The Real Projective Plane Chapter 9. Models for non-Euclidian Geometry 9.1 Three Types of Geometry 9.2 Hyperbolic Geometry 9.3 Elliptic Geometry 9.4 Euclidian and non-Euclidian Geometry in Space 9.5 Riemannian Geometry Chapter 10. Making Things Precise 10.1 Relations and Their Uses 10.2 Identification of Points 10.3 Our Number System 10.3.1 The integers 10.3.2 The rational numbers 10.3.3 The real numbers 10.3.4 The complex numbers Chapter 11. Projective Space 11.1 Coordinates in the Projective Plane 11.2 Projective n-Space 11.3 Affine and Projective Coordinate Systems Chapter 12. Geometry in the Affine and the Projective plane 12.1 The Theorem of Desargues 12.2 Duality for the projective plane 12.3 Naive Definition and First Examples of Affine Plane Curves 12.4 Straight Lines 12.5 Conic Sections in the Affine plane 12.6 Constructing Points on Conic Sections by Compass and Straightedge 12.7 Further Properties of Conic Sections 12.8 Conic Sections in the Projective Plane 12.9 The Theorems of Pappus and Pascal Chapter 13. Algebraic Curves of Higher Degrees in the Affine Plane 13.1 The Cubical Curves in the affine plane 13.1.1 Cubical parabolas 13.1.2 Semi cubical parabolas 13.1.3 Degenerate curves and degeneration of a family of curves 13.1.4 Folium of Descartes 13.1.5 Elliptic curves 13.1.6 Trisectrix of MacLaurin and the Clover leaf curve 13.2 Curves of Degree Higher than Three 13.3 Affine Algebraic Curves 13.4 Singularities and Multiplicities 13.5 Tangency Chapter 14. Higher Geometry in the Projective plane 14.1 Projective Curves 14.2 Projective Closure and Affine Restriction 14.3 Smooth and Singular Points on Affine and Projective Curves 14.4 The Tangent of a Projective Curve 14.5 Projective Equivalence 14.6 Asymptotes 14.7 General Conchoids 14.8 The Dual Curve 14.9 The Dual of Pappus'Theorem 14.10 Pascals Mysterium Hexagrammicum Chapter 15. Sharpening the Sword of Algebra 15.1 On Rational Polynomials 15.2 The Minimal Polynomial 15.3 The Euclidian Algorithm 15.4 Number Fields and Field Extensions 15.5 More on Field Extensions Chapter 16. Constructions with Straightedge and Compass 16.1 Review of Legal Constructions 16.2 Constructible Points 16.3 What is Possible? 16.4 Trisecting an Angle 16.5 Doubling the Cube 16.6 Squaring the Circle 16.7 Regular Polygons 16.8 Constructions by Folding 16.9 Concluding Remarks on Constructions Chapter 17. Fractal Geometry 17.1 Fractals and their Dimensions 17.2 The von Koch Snowflake Curve 17.3 Fractal Shapes in Nature 17.4 The Sierpinski Triangles 17.5 A Cantor Set Chapter 18. Catastrophe Theory 18.1 The Cusp Catastrophe: Geometry of a Cubic Surface 18.2 Rudiments of Control Theory

Detecting Algebraic Curves in Bad Position

Two affine algebraic curves over the complex numbers are in bad position if they have at least two distinct common points on some vertical line, i.e. for some α∈C there are distinctβ1 , β2∈C such that (α,β1 ) and (α, β2) are common points of the curves. We describe a method for detecting this situation and use it to develop an algorithm for finding the common points of two algebraic curves along with their multiplicities. We also give a worst case runtime analysis for the two algorithms and include a brief comparison of the second one with an approach using Gröbner bases.

Finding a Basis of a Linear System with Pairwise Distinct Discrete Valuations on an Algebraic Curve

Under the assumption that we have defining equations of an affine algebraic curve in special position with respect to a rational place Q, we propose an algorithm computing a basis of L(D) of a divisor D from an ideal basis of the ideal L(D+∞Q) of the affine coordinate ring L(∞Q) of the given algebraic curve, where L(D+∞Q) :=∪i=1∞L(D+iQ). Elements in the basis produced by our algorithm have pairwise distinct discrete valuations at Q, which is convenient in the construction of algebraic geometry codes. Our method is applicable to a curve embedded in an affine space of arbitrary dimension, and involves only the Gaussian elimination and the division of polynomials by the Gröbner basis for the ideal defining the curve.

Operator formalism on general algebraic curves

The usual Laurent expansion of the analytic tensors on the complex plane is generalized to any closed and orientable Riemann surface represented as an affine algebraic curve. As an application, the operator formalism for the b − c systems is developed. The physical states are expressed by means of creation and annihilation operators as in the complex plane and the correlation functions are evaluated starting from simple normal ordering rules. The Hilbert space of the theory exhibits an interesting internal structure, being splitted into n ( n is the number of branches of the curve) independent Hilbert spaces. In this way we are able to realize new kinds of conformal field theories at genus zero with symmetry group Vir n ⊗ G, Vir being the Virasoro group and G denoting a discrete and nonabelian crystallographic group. Exploiting the operator formalism a large collection of explicit formulas of string theory is derived. Finally, we develop as an important byproduct new methods in order to handle differential equations related to monodromy, like the Riemann monodromy problem.

Asymptotes and centers of affine algebraic curves

In the \(18^{{\rm th}}\) and \(19^{{\rm th}}\) century many beautiful theorems about the intersection of plane algebraic curves have been discovered. The book of Coolidge [C] contains a large collection of such results. One may ask which of these theorems can be generalized to curves in higher dimensional spaces. In this note we wish to discuss a generalization of a theorem of Waring and apply it to the question which affine algebraic curves have a unique "center".

Plane curves associated to character varieties of 3-manifolds

Consider a compact 3-manifold M with boundary consisting of a single torus. The papers [CS1, CS2, CGLS] discuss the variety of characters of SL2(C) representations of zl(M), and some of the ways in which the topological structure of M is reflected in the algebraic geometry of the character variety. We will describe in this paper a certain affine algebraic curve DM which is naturally associated to the character variety of M. A basis B = {~, 9.1/} for the peripheral subgroup of M determines an embedding P8 of DM into C* x C* with coordinates l and m. The closure in C 2 of ps(DM) is a plane algebraic curve and therefore is defined by a polynomial AM,n0, m) that, after certain normalizations, is uniquely determined up to multiplication by constants. The polynomial AM, B is effectively computable and is an invariant of the manifold M together with the choice of basis B. The results in this paper describe how geometric properties of the character variety, and hence topological properties of M, are reflected by the polynomial AM, B. In the case that M is the complement of a knot K in a homology 3-sphere we may take the basis B to consist of the longitude and meridian of K. With the usual orientation conventions this basis is well-defined modulo the involution which inverts both the longitude and meridian. It will follow from the construction of DM that if the basis B' is obtained from B by inverting both generators then the regular maps PB and PB, have the same image. Thus Ar:= AM, s = AM, B, is an invariant of the knot. The polynomial AM, B displays, in a striking way, information about the incompressible surfaces in M. This involves the Newton polygon of AM.B,

Siegel’s theorem for complex function fields

We give a short proof of the finiteness of the set of integral points on an affine algebraic curve of genus at least one, defined over a function field of characteristic zero.

Complex algebraic plane curves via their links at infinity

Considering that the study of plane cuves has an over 2000 year history and is the seed from which modern algebraic geometry grew, surprisingly little is known about the topology of affine algebraic plane curves. We topologically classify “regular” algebraic plane curves in complex affine 2-space using “splice diagrams:” certain decorated trees that code Puiseux data at infinity. (The regularity condition — that the curve be a “typical” fiber of its defining polynomial — can conjecturally be avoided.) We also show that the splice diagram determines such algebraic information as the minimal degree of the curve, even in the irregular case. Among other things, this enables algebraic classification of regular algebraic plane curves with given topology.

Differential Operators on an Affine Curve

Let X denote an irreducible affine algebraic curve over an algebraically closed field k of characteristic zero. Denote by Dx the sheaf of differential operators on X, and D(X)=Γ(X,Dx), the ring of global differential operators on X. The following is established: THEOREM. D(X) is a finitely generated k-algebra, and a noetherian ring. Furthermore, D(X) has a unique minimal non-zero ideal J, and D(X)/J is a finite-dimensional k-algebra. Let X ˜ denoted the normalisation of X, and π: X ˜ →X the projection map. The main technique is to compare D(X) with D( X ˜ ). THEOREM. The following are equivalent: (i) π is injective, (ii) D(X) is a simple ring, (iii) D(X) is Morita equivalent to D( X ˜ ), (iv) the categories DX-Mod and D X ˜ -Mod are equivalent, (v) gr D(X) is noetherian, (vi) the global homological dimension of D(X) is 1. For higher-dimensional varieties the techniques produce examples of varieties X for which D(X) is right but not left noetherian.

Affine curves on which every point is a set-theoretic complete intersection

We consider here those reduced and irreducible affine algebraic curves on which every point is the set-theoretic complete intersection of the given curve and a hypersurface. These curves, for historical reasons explained in Section 1, are called “prefactorial.” Our joint research in this area stems from our common interest in a certain result due to Mm-thy and Pedrini 1121: over algebraically closed ground fields of characteristic 0, prefactorial curves are necessarily rational and nollsingular. Presented here are the results of our efforts to understand prefactoriaIity in the absence of “algebraically closed” and/or “characteristic 0.” Our methods are mainly algebraic, depending largely on an analysis of the ‘grayer-Vietoris sequence of the conductors’ (as in 112)) and on certain “well-known”’ arithmetic-geometric facts which we learned from a paper of Rosen [16j (and which seem not to be so readily avaitable elsewhere in the literature). It is also of interest to examine pr~fa~toriality from the viewpoint of the classical theory of aigebraic curves: Section 4 contains some of the product of our tllinking along those lines.

Automatic analysis of real algebraic curves

We describe a computer program for the automatic analysis of a real plane affine algebraic curve. The input to the program is a bivariate integral polynomial F ( x, y ); the outputs are a report on the real curve defined by F ( x, y ) = 0, and a picture of the curve. The report contains the following information: whether the curve is irreducible, whether singular, and whether bounded; the number of its connected components and the dimension of each; the number of singular, turning, and level points of the curve. Approximations to these special points can be obtained to any desired precision; the more precision, the more time required. The exact form of the picture is controlled by the user; a topologically correct but "linearized" picture can be produced relatively quickly, while a more accurate drawing can be generated but requires more time. The program makes essential use of the clustering cylindrical algebraic decomposition algorithm [Arnon DS: Algorithms for the geometry of semi-algebraic sets (Dissertation). Technical Report #436, Computer Science Department, University of Wisconsin-Madison, 1981].

The Lefshetz vanishing cycles on a projective nonsingular plane curve

We deal in this paper with an explicit construction of Lefschetz vanishing cycles, which arise from a pencil of plane algebraic curves. Our main goal is Theorem 1, which helps to prove the Topological Decomposit ion Theorem for algebraic surfaces in CP 3 (Mandelbaum and Moishezon [7]). We consider a polynomial equation F(x,y)=O, which defines an affine plane algebraic curve C. We assume that the closure of C in CP 2 is nonsingular, and not tangent to the line at infinity. We consider a pencil of curves V~ :F(x, y) =s. There is a finite set Z of critical values of s, such that V~ has singularities. The noncritical values of s are called regular values. By our assumptions s = 0 is a regular value. Let c be a critical value, such that ~ possesses only one nondegenerate singular point P. Then e is called a simple critical value. We can choose analytic local coordinates u(x, y), v(x,y) in a neighbourhood U of P in the plane, such that F(x, y) = c + u(x, y)2 + v(x, y)2 in U and (u, v) (P) = (0, 0). We can also choose a neighbourhood D of c in the parameter plane, such that Vsc~ U has a 1-sphere as a deformation retract for each seD. This sphere represents the Lefschetz vanishing cycle on Vs corresponding to c. We will choose a special representation of the Lefschetz cycle. Let ),(t), 0 < t < 1, be a simple path in the parameter plane, such that 7(0)=0, ?(1)=c and ?(t) is not a critical value for t # 1. Then there exists a continuous deformation I,: Vo~ Vm),