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2008•arXiv (Cornell University)Open access

Algebraic Curves Soluble by Radicals

J. Rafael Sendra, David Sevilla

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Abstract

We present the notion of algebraic curve soluble (or parametrizable) by radicals. We prove that every irreducible curve (not necessarily plane) of genus less o equal to 4 is soluble by radicals. Moreover, we provide algorithms for finding radical parametrizations in those cases. In addition, irreducible plane curves of degree d and having at least a point of multiplicity d − r, with 1 ≤ r ≤ 4, are shown to be soluble by radicals. As a consequence, every irreducible plane curve of degree less or equal to 5 is soluble by radicals; similarly if the irreducible curve is singular of degree 6. 1

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We present the notion of algebraic curve soluble (or parametrizable) by radicals. We prove that every irreducible curve (not necessarily plane) of genus less o equal to 4 is soluble by radicals. Moreover, we provide algorithms for finding radical parametrizations in those cases. In addition, irreducible plane curves of degree d and having at least a point of multiplicity d − r, with 1 ≤ r ≤ 4, are shown to be soluble by radicals. As a consequence, every irreducible plane curve of degree less or equal to 5 is soluble by radicals; similarly if the irreducible curve is singular of degree 6. 1

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Available abstract

We present the notion of algebraic curve soluble (or parametrizable) by radicals. We prove that every irreducible curve (not necessarily plane) of genus less o equal to 4 is soluble by radicals. Moreover, we provide algorithms for finding radical parametrizations in those cases. In addition, irreducible plane curves of degree d and having at least a point of multiplicity d − r, with 1 ≤ r ≤ 4, are shown to be soluble by radicals. As a consequence, every irreducible plane curve of degree less or equal to 5 is soluble by radicals; similarly if the irreducible curve is singular of degree 6. 1

Key concepts: Plane curve, Multiplicity (mathematics), Algebraically closed field, Algebraic curve, Mathematics, Degree (music), Plane (geometry), Zero (linguistics)

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