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Nicholas J. Willis

Publications and source records attributed to Nicholas J. Willis.

5 recordsLinked to original sources
arXiv2019

Singular Points of High Multiplicity for Septic Curves

For real irreducible algebraic curves of the seventh degree, there are 22 types of singular points of multiplicity six, 174 types of singular points of multiplicity five, and at least 182 types of singular points of multiplicity four. For complex irreducible algebraic curves of the seventh degree, there are 12 types of singular points of multiplicity six, 92 types of singular points of multiplicity five, and at least 92 types of singular points of multiplicity four. In the final section of the paper, a wide variety of open problems on the classification of singular points of plane algebraic cuves is explicitly described.

math.AG↗
arXiv2008

How to Compute a Puiseux Expansion

In this paper, an explanation of the Newton-Peiseux algorithm is given. This explanation is supplemented with well-worked and explained examples of how to use the algorithm to find fractional power series expansions for all branches of a polynomial at the origin.

math.AG↗
arXiv2008

Singular Points of Real Quintic Curves Via Computer Algebra

There are 42 types of real singular points for irreducible real quintic curves and 49 types of real singular points for reducible real quintic curves. The classification of real singular points for irreducible real quintic curves is originally due to Golubina and Tai. There are 28 types of singular points for irreducible complex quintic curves and 33 types of singular points for reducible complex quintic curves. We derive the complete classification with proof by using the computer algebra system Maple. We clarify that the classification is based on computing just enough of the Puiseux expansion to separate the branches. Thus, a major component of the proof consists of a sequence of large symbolic computations that can be done nicely using Maple.

math.AG↗
arXiv2007

Singular points of real quartic curves via computer algebra

There are thirteen types of singular points for irreducible real quartic curves and seventeen types of singular points for reducible real quartic curves. This classification is originally due to D.A. Gudkov. There are nine types of singular points for irreducible complex quartic curves and ten types of singular points for reducible complex quartic curves. We derive the complete classification with proof by using the computer algebra system Maple. We clarify that the classification is based on computing just enough of the Puiseux expansion to separate the branches. Thus, the proof consists of a sequence of large symbolic computations that can be done nicely using Maple.

math.AG↗