Browsing by Subject "covering radius"

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Access status: Open Access ,
Geometric properties of the lattice of polynomials with integer coefficients
(Wydawnictwa AGH, 2024) Lipnicki, Artur; Śmietański, Marek J.
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 $\boldsymbol{P}_n \cap \boldsymbol{M}_r$.
Access status: Open Access ,
Uniform approximation by polynomials with integer coefficients
(2016) Lipnicki, Artur
Let $r$, $n$ be positive integers with $n\ge 6r$. Let $P$ be a polynomial of degree at most n on $[0,1]$ with real coefficients, such that $P^{(k)}(0)/k!$ and $P^{(k)}(1)/k!$ are integers for $k=0,\dots,r-1$. It is proved that there is a polynomial $Q$ of degree at most $n$ with integer coefficients such that $|P(x)-Q(x)|\le C_1C_2^r r^{2r+1/2}n^{-2r}$ for $x\in[0,1]$, where $C_1$, $C_2$ are some numerical constants. The result is the best possible up to the constants.