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.
- W. Banaszczyk, A. Lipnicki, On the lattice of polynomials with integer coefficients: the covering radius in \(L_p(0,1)\), Ann. Polon. Math. 121 (2015), 123-144. https://doi.org/10.4064/ap115-2-2
- R.L. Burden, J.D. Faires, A.M. Burden Numerical Analysis, 10th Edition, Cengage Learning, 2015.
- L.B.O. Ferguson, Approximation by Polynomials with Integer Coefficients, Amer. Math. Society, Providence, R.I., 1980.
- L.B.O. Ferguson, What can be approximated by polynomials with integer coefficients, Amer. Math. Monthly 113 (2006), 403-414.
- L.V. Kantorowič, Quelques observations sur l'approximation de fonctions au moyen de polynômes à coefficients entiers, Izv. Akad. Nauk SSSR Ser. Mat. 9 (1931), 1163-1168.
- A. Lipnicki, Uniform approximation by polynomials with integer coefficients, Opuscula Math. 36 (2016), no. 4, 489-498. https://doi.org/10.7494/opmath.2016.36.4.489
- Y. Okada, On approximate polynomials with integral coefficients only, Tohoku Math. J. 23 (1923), 26-35.
- J. Pál, Zwei kleine Bemerkungen, Tohoku Math. J. 6 (1914), 42-43.
- R.M. Trigub, Approximation of functions by polynomials with integer coefficients, Izv. Akad. Nauk SSSR Ser. Mat. 26:2 (1962), 261-280.
- Artur Lipnicki
https://orcid.org/0000-0001-9848-6157- University of Lodz, Faculty of Mathematics and Computer Science, Banacha 22, 90-238 Łódź, Poland
- Marek J. Śmietański (corresponding author)
- https://orcid.org/0000-0002-6557-6436
- University of Lodz, Faculty of Mathematics and Computer Science, Banacha 22, 90-238 Łódź, Poland
- Communicated by Palle E.T. Jorgensen.
- Received: 2023-12-18.
- Revised: 2024-01-26.
- Accepted: 2024-01-27.
- Published online: 2024-04-29.
