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Gaussian Decomposition of Real Plane Algebraic Curves

Barry H. Dayton

Wolfram Technology Conference 2015
Conference location

Champaign, Illinois USA

In 1799 Carl Friedrich Gauss gave a proof of the Fundamental Theorem of Algebra using an implied decomposition theorem for real algebraic plane curves. This decomposition was made more explicit in his 1849 fourth proof. In this talk we give a modern version of his 1849 decomposition theorem and a black box Mathematica algorithm which, in many cases, can decompose a curve in accordance with Gauss' Theorem. This decomposition is related to oval decomposition for non-singular curves but differs for singular curves. An advantage is that, in principle, the algorithm can handle all singularities, although possibly in non-black box mode. We work in the Gaussian, (affine) plane. In addition to decomposing the curve there is also a decomposition of the complement of the curve into connected components. The algorithm produces a data object consisting of bounding triangles for a real plane algebraic curve. This object can be used to graph the curve, graph its complement, transform the curve, or to answer the identification question, “find the component of the curve containing a specified point”. The decomposition algorithm has been tested on several large data bases of curves including Gauss Curves, Newton Hyperbolas and Rational Curves.

*Mathematics > Algebra
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1445102309.nb (102 KB) - Mathematica Notebook
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