On the lattice of polynomials with integer coefficients: successive minima in
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.
