INTRINSIC PROPERTIES AND INVARIANT OF PLANAR
ALGEBRAIC CURVES AND THEIR APPLICATIONS
ZHONGXUAN LUO
abstract
Algebraic curve is an important research object and tool in many mathematical
fields such as pure mathematics (algebraic geometry, transcendental numbers) and
applied mathematics(interpolations and splines, CAGD, and approximation, etc.).
The aim of this talk is to show an equivalence relation between the study of the
intrinsic properties of planar algebraic curves and the study of singularity of bivari-
ate spline spaces, and to investigate the absorbing properties of algebraic curves
by means of spline approach. To study the intrinsic properties of algebraic curves
in the projection plane, some definitions such as characteristic ratio of points on a
line, characteristic mapping of point or line, and characteristic number of algebraic
curve are proposed, and then an intrinsic invariant of any algebraic curve in a so
called Pascal space is discovered by using the above concepts and the spline method.
As another main result, the classical Pascal’s theorem is generalized skillfully for
the case of algebraic curve of higher degree n ≥ 3 in the Pascal space by recursive
form. This extension is quite different from the generalizations to Pascal’s theorem
such as Chasles’s Theorem and Cayley-Bacharach Theorem in algebraic geometry.
Some examples to the Pascal’s type theorem are given to interpret our results more
clearly. Meanwhile, the singularity of the spline space over Morgan-Scott’s parti-
tion can be settled from our results by using the duality principle. Our results seem
to be subservient to further study the classification and rational parametrization of
algebraic curves, and classification of finite groups, etc.
Department of Applied Mathematics, Dalian University of Technology, Dalian, 116024,
China
E-mail address: luozx9609@vip.sina.com
Date: May 10, 2006.
2000 Mathematics Subject Classification. Primary 14H50; Secondary 65D07; 41A15.
Key words and phrases. Planar algebraic curve, singularity of spline space, characteristic ratio,
characteristic mapping, characteristic number, Pascal curve.
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