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Representation of squares by nonsingular cubic forms

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Abstract

We prove an asymptotic formula for the number of representations of squares by nonsingular cubic forms in six or more variables. The main ingredients of the proof are Heath-Brown’s form of the Circle Method and various exponential sum results. The depth of the exponential sum results is comparable to Hooley’s work on cubic forms in nine variables, in particular we prove an analogue of Katz’ bound.

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Authors and Affiliations

  1. Mathematisch Institut, Universiteit Utrecht Hans Freudenthalgebouw, Budapestlaan 6, 3584 CD, Utrecht, The Netherlands

    Lasse Grimmelt

  2. Department of Mathematics, Columbia University, 2990 Broadway, New York, NY, 10027, USA

    Will Sawin

Authors
  1. Lasse Grimmelt
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  2. Will Sawin
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Correspondence to Lasse Grimmelt.

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Grimmelt, L., Sawin, W. Representation of squares by nonsingular cubic forms. Isr. J. Math. 242, 501–547 (2021). https://doi.org/10.1007/s11856-021-2116-2

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  • DOI: https://doi.org/10.1007/s11856-021-2116-2

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