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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.

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.