Affine Plane Curves Research Articles

Construction of affine plane curves with one place at infinity

Construction of affine plane curves with one place at infinity

The Orevkov invariant of an affine plane curve

We show that although the fundamental group of the complement of an algebraic affine plane curve is not easy to compute, it possesses a more accessible quotient, which we call the Orevkov invariant.

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

Abhyankar-Sathaye embedding problem in dimension three

Abhyankar-Moh and Suzuki proved that if an irreducible polynomial f ∈ C[x1,x2] in two complex variables x1 and x2 defines the affine plane curve C =( f =0 )⊂ A 2 , which is isomorphic to the affine line: C ∼ A 1 ,t henf is a variable of C[x1,x2], i.e., there exists a polynomial g ∈ C[x1,x2] such that C[f,g ]= C[x1,x2] (cf. [A-M75], [Su74]). In this article, we prove under some additional assumptions that the similar result holds in the three-dimensional case, namely, if an irreducible polynomial f ∈ C[x1,x2,x3] in three complex variables x1,x2 and x3 defines the hypersurface S =( f =0 )⊂ A 3 , which is isomorphic to the affine plane: S ∼ A 2 ,t henf is a variable of C[x1,x2,x3], i.e., there are polynomials g,h ∈ C[x1,x2,x3] such that C[f,g,h ]= C[x1,x2,x3]. Moreover, we shall determine the detailed form of such a polynomial f ∈ C[x1,x2,x3] for the special case.

Irregular links at infinity of complex affine plane curves

Journal Article Irregular links at infinity of complex affine plane curves Get access WD Neumann WD Neumann Department of Mathematics, The University of Melbourne, Parkville, Vic 3052, Australia. E-mail: neumann@maths.mu.oz.au Search for other works by this author on: Oxford Academic Google Scholar The Quarterly Journal of Mathematics, Volume 50, Issue 199, September 1999, Pages 301–320, https://doi.org/10.1093/qjmath/50.199.301 Published: 01 September 1999

Irreducible plane curves with the Albanese dimension 2

Let B B be a plane curve given by an equation F ( X 0 , X 1 , X 2 ) = 0 F(X_{0}, X_{1}, X_{2}) = 0 , and let B a B_{a} be the affine plane curve given by f ( x , y ) = F ( 1 , x , y ) = 0 f(x, y) = F(1,x, y) = 0 . Let S n S_{n} denote a cyclic covering of P 2 {\mathbf {P}}^{2} determined by z n = f ( x , y ) z^{n} = f(x, y) . The number max n ∈ N ( dim ⁡ ℑ ( S n → Alb ⁡ ( S n ) ) ) \max _{ n \in {\mathbf {N}}} \left ( \operatorname {dim} \Im (S_{n} \to \operatorname {Alb} (S_{n})) \right ) is called the Albanese dimension of B a B_{a} . In this article, we shall give examples of B a B_{a} with the Albanese dimension 2.

Affine plane curves with one place at infinity

In this paper we give a new algebro-geometric proof to the semi-group theorem due to Abhyankar-Moh for the affine plane curves with one place at infinity and its inverse theorem due to Sathaye-Stenerson. The relations between various invariants of these curves are also explained geometrically. Our new proof gives an algorithm to classify the affine plane curves with one place at infinity with given genus by computer.

Topology of families of affine plane curves

We determine bifurcation sets of families of affine curves and study the topology of such families.

Affine plane curve evolution: a fully consistent scheme

We present an accurate numerical scheme for the affine plane curve evolution and its morphological extension to grey-level images. This scheme is based on the iteration of a nonlocal, fully affine invariant and numerically stable operator, which can be exactly computed on polygons. The properties of this operator ensure that a few iterations are sufficient to achieve a very good accuracy, unlike classical finite difference schemes that generally require a lot of iterations. Convergence results are provided, as well as theoretical examples and experiments.

Remark on the equisingularity of families of affine plane curves

Remark on the equisingularity of families of affine plane curves

Geometry of plane curves via Tschirnhausen resolution tower

We show the existence of toric resolution tower for an irreducible curve singularity which is explicitly described by Tschirnhausen polynomials. We deduce for a smooth affine plane curve from its topology restrictions for its singularity at infinity. For instance, we discribe the singularities at infinity (up to equisingular deformation) for curves of genus 0,1 and 2. From this follows an extension of the Abhyankar-Moh-Suzuki theorem to genus 1 and 2.

Uniqueness of plane embeddings of special curves

For a family of special affine plane curves, it is shown that their embeddings in the affine plane are unique up to automorphisms of the affine plane. Examples are also given for which the embedding is not unique. We also discuss the Lin-Zaidenberg estimate of the number of singular points of an irreducible curve in terms of its rank. Formulas concerning the rank of the curve lead to an alternate simpler version of the proof of the Epimorphism Theorem.

A simple approach for construction of algebraic-geometric codes from affine plane curves

The current algebraic-geometric (AG) codes are based on the theory of algebraic-geometric curves. In this paper we present a simple approach for the construction of AG codes, which does not require an extensive background in algebraic geometry. Given an affine plane irreducible curve and its set of all rational points, we can find a sequence of monomials x/sup i/y/sup j/ based on the equation of the curve. Using the first r monomials as a basis for the dual code of a linear code, the designed minimum distance d of the linear code, called the AG code, can be easily determined. For these codes, we show a fast decoding procedure with a complexity O(n/sup 7/3/), which can correct errors up to [(d-1/2]. For this approach it is neither necessary to know the genus of curve nor the basis of a differential form. This approach can be easily understood by most engineers.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Algebraic geometric codes on certain plane curves

AbstractNecessary and sufficient conditions are given for projective plane curves to have cusps isomorphic to the origin of an affine plane curve xa + yb. Based on them, a family of curves Cab having only one such cusp as a singular point and a family of curves rCab having only two such cusps as singular points are formulated. Also, the structures of algebraic geometric codes generated on curves contained in these families are shown, where a and b are relatively prime. Then the genus is (a − 1)/(b − 1)/2 and the basis of linear space L(m · P) is given by rational functions xiyi only. Second, by giving concrete examples, it is shown that families of curves Cab and rCab contain many curves which attain Hasse‐Weil upper bound.

Construction of codes on arbitrary plane curves

In 1981, Goppa established an epoch-making method for constructing a code utilizing the theory of algebraic curves [1]. This is called the algebraic curve code. This paper develops the theory on an arbitrary curve, which may contain singular points and discusses the method of constructing the code. The method employed in this paper is to eliminate the singular point by applying the transformation called blowing-up and to construct the code on the nonsingular model. The number of genes of curves can be derived simultaneously. It is made easy to derive the bases of the linear spaces L(G) and Ω(G-D), as in the case of the nonsingular curve. The algebraic curve C considered in this paper is the irreducible projected plane curve or the algebraic curve which is obtained by gluing irreducible affine plane curves. In this case, the nonsingular model of C can also be represented as a gluing of irreducible affine plane curves. Then the method is applied to the hyperelliptic plane curve on which the number of rational points achieves the Hasse-Weil bound or certain hypernonelliptic plane curves. It is shown that the precise structure of the code can be examined.

Relative Frobenius of plane singularities

In view of the well-known conjecture concerning the classification of lines in the affine plane in characteristic p > 0, it is desirable to understand how the characteristic pairs of an irreducible algebroid plane curve are affected by the Frobenius. This paper determines the relation between the char- acteristic sequences (x, y) and (x, f), where x and y are formal power series in one variable with coefficients in a field of characteristic p > 0 . Throughout this paper, k is an algebraically closed field of characteristic p > 0, t is an indeterminate over k and k((t)) is the field of fractions of the ring k((t)) of formal power series in t. An affine plane curve with one place at infinity may be thought of as an epimorphism e: k(X, Y) -> A, where A is a one dimensional domain contained in all but one (say A g R ) valuation rings of qt(A)/k. Note that if x = e(X) and y = e(Y) then e':k(X, Y) -, A1 = k(x, yp) defined by e'(X) = x and e'(Y) = yp is also an affine plane curve with one place at infinity. We say that e is obtained from e by performing the relative Frobenius operation. This paper is part of an attempt to answer: Question 1. How is the singularity at infinity affected by the Frobenius? As a motivation for this question we recall the well-known conjecture which asserts that every affine plane curve biregular to the affine line can be obtained from a line of degree one by performing a finite sequence of operations, each operation being either an automoprhism of the plane or the Frobenius, the latter being allowed only if A = A1. (See (1, 2, 4, 5).) Let the notation be as above. Then the completion of R may be identified with k((i)) and as a result x, y £ k((t)) have negative orders (unless x £ k or y £ k, which is a trivial case). Define x^, yoo£ tk((t\) by ' (1/x, y/x), if ordx < ordy,

Local trees in the theory of affine plane curves

Local trees in the theory of affine plane curves

Plane Algebraic Curves: An Introduction via Valuations.

1. Prerequisites 2. Some Facts About Polynomials 3. Affine Plane Curves 4. Tangent Spaces 5. The Local Ring at a Point 6. Projective Plane Curves 7. Rational Mappings, Birational Correspondences and Isomorphisms of Curves 8. Examples of Rational Curves 9. The Correspondence between Valuations and Points 10. An Overview and Sideways Glance 11. Divisors 12. The Divisor of a Function Has Degree 13. Riemann's Theorem 14. The Genus of a Nonsingular Plane Curve 15. Curves of Genus 0 and 1 16. A Classification of Isomorphism Classes of Curves of Genus 1 17. The Genus of a Singular Curve 18. Inflection Points on Plane Curves 19. Bezout's Theorem 20. Addition on a Nonsingular Cubic 21. Derivations, Differentials and the Canonical Class 22. Adeles and the Riemann-Roch Theorem