1st Edition
Pencils of Cubics and Algebraic Curves in the Real Projective Plane
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.
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
Biography
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.