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Geometric properties of the lattice of polynomials with integer coefficients

Lipnicki Artur, Śmietański Marek J.

Opuscula Mathematica

2024

Vol. 44, no. 4

565--585

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.

On some classes of block repetition codes with covering radius of the codes Z32

Pandian P. Chella

Journal of Mathematics and Applications

2023

Vol. 46

69--77

In this paper, to obtain the bounds for some classes of repetition codes with covering radius by using various weight and also the same size and different size of length in repetition codes over a finite ring Z32 .

Uniform approximation by polynomials with integer coefficients

Lipnicki A.

Opuscula Mathematica

2016

Vol. 36, no. 4

489--498

Let r, n be positive integers with n ≥ 6r. Let P be a polynomial of degree at most n on [0,1] with real coefficients, such that [formula] are integers for k = 0,…, r — 1. It is proved that there is a polynomial Q of degree at most n with integer coefficients such that [formula] for x ∈ [0,1], where C1, C2 are some numerical constants. The result is the best possible up to the constants.