Lattices over Polynomial Rings and Applications to Function Fields

2016/01/07 by Bauch, Jens-Dietrich · 1 citation
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1601.01361

Abstract

This paper deals with lattices (L,\Vert~\Vert) over polynomial rings, where L is a finitely generated module over k[t], the polynomial ring over the field k in the indeterminate t, and \Vert~\Vert is a discrete real-valued length function on L⊗_k[t]k(t). A reduced basis of (L,\Vert~\Vert) is a basis of L whose vectors attain the successive minima of (L,\Vert~\Vert). We develop an algorithm which transforms any basis of L into a reduced basis of (L,\Vert~\Vert). By identifying a divisor D of an algebraic function field with a lattice (L,\Vert~\Vert) over a polynomial ring, this reduction algorithm can be addressed to the computation of the Riemann-Roch space of D and the successive minima of (L,\Vert~\Vert), without the use of any series expansion.

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