On the lattice of polynomials with integer coefficients: successive minima in $L_2(0,1)$
Volume 124 / 2020
Annales Polonici Mathematici 124 (2020), 109-128
MSC: Primary 41A10; Secondary 52C07.
DOI: 10.4064/ap190413-20-10
Published online: 21 February 2020
Abstract
Let $\boldsymbol{P} _n^\mathbb{Z} $ be the additive subgroup of the real Hilbert space $L_2(0,1)$ consisting of polynomials of order $\le n$ with integer coefficients. We may treat $\boldsymbol{P} _n^\mathbb{Z} $ as a lattice in $(n+1)$-dimensional Euclidean space; let $\lambda _i(\boldsymbol{P} _n^\mathbb{Z} )$ ($1\le i\le n+1$) be the corresponding successive minima. We give rather precise estimates of $\lambda _i(\boldsymbol{P} _n^\mathbb{Z} )$ for $i\gtrsim \frac 23n$.
