Opuscula Math. 44, no. 4 (2024), 565-585
https://doi.org/10.7494/OpMath.2024.44.4.565

 
Opuscula Mathematica

Geometric properties of the lattice of polynomials with integer coefficients

Artur Lipnicki
Marek J. Śmietański

Abstract. This paper is related to the classic but still being examined issue of approximation of functions by polynomials with integer coefficients. Let \(r\), \(n\) be positive integers with \(n \ge 6r\). Let \(\boldsymbol{P}_n \cap \boldsymbol{M}_r\) be the space of polynomials of degree at most \(n\) on \([0,1]\) with integer coefficients such that \(P^{(k)}(0)/k!\) and \(P^{(k)}(1)/k!\) are integers for \(k=0,\dots,r-1\) and let \(\boldsymbol{P}_n^\mathbb{Z} \cap \boldsymbol{M}_r\) be the additive group of polynomials with integer coefficients. We explore the problem of estimating the minimal distance of elements of \(\boldsymbol{P}_n^\mathbb{Z} \cap \boldsymbol{M}_r\) from \(\boldsymbol{P}_n \cap \boldsymbol{M}_r\) in \(L_2(0,1)\). We give rather precise quantitative estimations for successive minima of \(\boldsymbol{P}_n^\mathbb{Z}\) in certain specific cases. At the end, we study properties of the shortest polynomials in some hyperplane in \(\boldsymbol{P}_n \cap \boldsymbol{M}_r\).

Keywords: approximation by polynomials with integer coefficients, lattice, covering radius, roots of polynomial.

Mathematics Subject Classification: 41A10, 52C07, 26C10, 65H04.

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  • Communicated by Palle E.T. Jorgensen.
  • Received: 2023-12-18.
  • Revised: 2024-01-26.
  • Accepted: 2024-01-27.
  • Published online: 2024-04-29.
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Cite this article as:
Artur Lipnicki, Marek J. Śmietański, Geometric properties of the lattice of polynomials with integer coefficients, Opuscula Math. 44, no. 4 (2024), 565-585, https://doi.org/10.7494/OpMath.2024.44.4.565

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