Representing integers by multilinear polynomials

Abstract

Let \(F(\varvec{x})\) be a homogeneous polynomial in \(n \ge 1\) variables of degree \(1 \le d \le n\) with integer coefficients so that its degree in every variable is equal to 1. We give some sufficient conditions on F to ensure that for every integer b there exists an integer vector \(\varvec{a}\) such that \(F(\varvec{a}) = b\). The conditions provided also guarantee that the vector \(\varvec{a}\) can be found in a finite number of steps.

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Authors' contributions

We thank Levent Alpoge and the anonymous referees for some very helpful remarks.

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Correspondence to Lenny Fukshansky.

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Fukshansky acknowledges support by Simons Foundation Grant #519058.

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Böttcher, A., Fukshansky, L. Representing integers by multilinear polynomials. Res. number theory 6, 38 (2020). https://doi.org/10.1007/s40993-020-00218-w

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Keywords

  • Polynomials
  • Integer representations
  • Unimodular matrices
  • Linear and multilinear forms

Mathematics Subject Classification

  • Primary 11D85
  • Secondary 11C08
  • 11C20
  • 11G50