TY  - JOUR
ID  - Lipnicki2024
LB  - Lipnicki2024
AU  - Lipnicki, Artur
AU  - Śmietański, Marek J.
TI  - Geometric properties of the lattice of polynomials with integer coefficients
ST  - Geometric properties of the lattice of polynomials with integer coefficients
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2024
VL  - 44
IS  - 4
SP  - 565
EP  - 585
DO  - https://doi.org/10.7494/OpMath.2024.44.4.565
UR  - https://doi.org/10.7494/OpMath.2024.44.4.565
AB  - 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 \(\boldsymbol{P}_n \cap \boldsymbol{M}_r\).
KW  - approximation by polynomials with integer coefficients
KW  - lattice
KW  - covering radius
KW  - roots of polynomial
SN  - 1232-9274
ER  -