On convolutions of algebraic curves

Authors:
Jan Vrek;Miroslav Lávika
Affiliations:
University of West Bohemia, Faculty of Applied Sciences, Department of Mathematics, Univerzitní 8, 301 00 Plze, Czech Republic;University of West Bohemia, Faculty of Applied Sciences, Department of Mathematics, Univerzitní 8, 301 00 Plze, Czech Republic
Venue:
Journal of Symbolic Computation
Year:
2010

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Abstract

We focus on the investigation of relations between plane algebraic curves and their convolution. Since the convolution of irreducible algebraic curves is not necessarily irreducible, an upper bound for the number of components is given. Then, a formula expressing the convolution degree using the algebraic degree and the genus of the curve is derived. In addition, a detailed analysis of the so-called special and degenerated components is discussed. We also present some special results for curves with low convolution degree and for rational curves, and use our results to investigate the relation with the theory of the classical offsets and Pythagorean Hodograph (PH) curves presented in Arrondo et al. (1997).