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On the lattice of polynomials with integer coefficients: successive minima in $L_2(0,1)$

Volume 124 / 2020

Wojciech Banaszczyk Annales Polonici Mathematici 124 (2020), 109-128 MSC: Primary 41A10; Secondary 52C07. DOI: 10.4064/ap190413-20-10 Published online: 21 February 2020


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 $i\gtrsim \frac 23n$.


  • Wojciech BanaszczykFaculty of Mathematics and Computer Science
    University of Lodz
    Banacha 22
    90-238 Łódź, Poland

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