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On the Length of Binary Forms

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Quadratic and Higher Degree Forms

Part of the book series: Developments in Mathematics ((DEVM,volume 31))

Abstract

The K-length of a form f in \(K[x_{1},\ldots,x_{n}]\), \(K \subset \mathbb{C}\), is the smallest number of d-th powers of linear forms of which f is a K-linear combination. We present many results, old and new, about K-length, mainly for n = 2, and often about the length of the same form over different fields. For example, the K-length of \(3{x}^{5} - 20{x}^{3}{y}^{2} + 10x{y}^{4}\) is three for \(K = \mathbb{Q}(\sqrt{-1})\), four for \(K = \mathbb{Q}(\sqrt{-2})\) and five for \(K = \mathbb{R}\).

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Authors and Affiliations

  1. Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL, 61801, USA

    Bruce Reznick

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  1. Bruce Reznick
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Editors and Affiliations

  1. Department of Mathematics, University of Florida Department of Mathematics, Gainesville, Florida, USA

    Krishnaswami Alladi

  2. Department of Mathematics, Princeton University, Princeton, New Jersey, USA

    Manjul Bhargava

  3. Department of Mathematics, University of Arizona, Tucson, Arizona, USA

    David Savitt

  4. Department of Mathematics, University of Arizona, Tuscon, Arizona, USA

    Pham Huu Tiep

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Reznick, B. (2013). On the Length of Binary Forms. In: Alladi, K., Bhargava, M., Savitt, D., Tiep, P. (eds) Quadratic and Higher Degree Forms. Developments in Mathematics, vol 31. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7488-3_8

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