Geometric properties of the lattice of polynomials with integer coefficients

• Autorzy: Lipnicki Artur , Śmietański Marek J.

Identyfikatory

DOI

10.7494/OpMath.2024.44.4.565

• Języki publikacji: EN

Abstrakty

EN

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 ≥ 6r. Let Pn ∩Mr 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, . . . , r − 1 and let PZn ∩Mr be the additive group of polynomials with integer coefficients. We explore the problem of estimating the minimal distance of elements of PZn ∩Mr from Pn ∩Mr in L2(0, 1). We give rather precise quantitative estimations for successive minima of PZn in certain specific cases. At the end, we study properties of the shortest polynomials in some hyperplane in Pn ∩Mr.

Słowa kluczowe

EN

approximation by polynomials with integer coefficients; lattice; covering radius; roots of polynomials

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2024
• Tom: Vol. 44, no. 4
• Strony: 565--585
• Opis fizyczny: Bibliogr. 9 poz., tab., wykr.

Twórcy

autor

Lipnicki Artur

• University of Lodz, Faculty of Mathematics and Computer Science, Banacha 22, 90–238 Łódź, Poland

• https://orcid.org/0000-0001-9848-6157

autor

Śmietański Marek J.

• University of Lodz, Faculty of Mathematics and Computer Science, Banacha 22, 90–238 Łódź, Poland

• https://orcid.org/0000-0002-6557-6436

Bibliografia

• [1] W. Banaszczyk, A. Lipnicki, On the lattice of polynomials with integer coefficients: the covering radius in Lp(0, 1), Ann. Polon. Math. 121 (2015), 123–144.
• [2] R.L. Burden, J.D. Faires, A.M. Burden, Numerical Analysis, 10th Edition, Cengage Learning, 2015.
• [3] L.B.O. Ferguson, Approximation by Polynomials with Integer Coefficients, Amer. Math. Society, Providence, R.I., 1980.
• [4] L.B.O. Ferguson, What can be approximated by polynomials with integer coefficients, Amer. Math. Monthly 113 (2006), 403–414.
• [5] 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.
• [6] A. Lipnicki, Uniform approximation by polynomials with integer coefficients, Opuscula Math. 36 (2016), no. 4, 489–498.
• [7] Y. Okada, On approximate polynomials with integral coefficients only, Tohoku Math. J. 23 (1923), 26–35.
• [8] J. Pál, Zwei kleine Bemerkungen, Tohoku Math. J. 6 (1914), 42–43.
• [9] R.M. Trigub, Approximation of functions by polynomials with integer coefficients, Izv. Akad. Nauk SSSR Ser. Mat. 26:2 (1962), 261–280.

Uwagi

Opracowanie rekordu ze środków MEiN, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2022-2023)