Geometric properties of the lattice of polynomials with integer coefficients

Abstract

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 $\boldsymbol{P}_n \cap \boldsymbol{M}_r$.