Pencils of Cubics and Algebraic Curves in the Real Projective Plane: 1st Edition (Paperback) book cover

Pencils of Cubics and Algebraic Curves in the Real Projective Plane

1st Edition

By Séverine Fiedler - Le Touzé

Chapman and Hall/CRC

226 pages | 107 B/W Illus.

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Description

Pencils of Cubics and Algebraic Curves in the Real Projective Plane thoroughly examines the combinatorial configurations of n generic points in RP². Especially how it is the data describing the mutual position of each point with respect to lines and conics passing through others.

The first section in this book answers questions such as, can one count the combinatorial configurations up to the action of the symmetric group? How are they pairwise connected via almost generic configurations? These questions are addressed using rational cubics and pencils of cubics for n = 6 and 7. The book’s second section deals with configurations of eight points in the convex position. Both the combinatorial configurations and combinatorial pencils are classified up to the action of the dihedral group D8. Finally, the third section contains plentiful applications and results around Hilbert’s sixteenth problem.

The author meticulously wrote this book based upon years of research devoted to the topic. The book is particularly useful for researchers and graduate students interested in topology, algebraic geometry and combinatorics.

Features:

  • Examines how the shape of pencils depends on the corresponding configurations of points
  • Includes topology of real algebraic curves
  • Contains numerous applications and results around Hilbert’s sixteenth problem

About the Author:

Séverine Fiedler-le Touzé has published several papers on this topic and has been invited to present at many conferences. She holds a Ph.D. from University Rennes1 and was a post-doc at the Mathematical Sciences Research Institute in Berkeley, California.

Table of Contents

Rational pencils of cubics and configurations of six or seven points in RP²

Points, lines and conics in the plane

Configurations of six points

Configurations of seven points

Pencils of cubics with eight base points lying in convex position in RP²

Pencils of cubics

List of conics

Link between lists and pencils

Pencils with reducible cubics

Classification of the pencils of cubics

Tabulars

Application to an interpolation problem

Algebraic curves

Hilbert’s 16th problem

M-curves of degree 9

M-curves of degree 9 with deep nests

M-curves of degree 9 with four or three nests

M-curves of degree 9 or 11 with non-empty oval

Curves of degree 11 with many nests

Totally real pencils of curves

About the Author

Séverine Fiedler-le Touzé has published several papers on this topic and has been invited to present at many conferences. She holds a Ph.D. from University Rennes1 and was a post-doc at the Mathematical Sciences Research Institute in Berkeley, California.

Subject Categories

BISAC Subject Codes/Headings:
MAT002000
MATHEMATICS / Algebra / General
MAT012000
MATHEMATICS / Geometry / General
MAT022000
MATHEMATICS / Number Theory