Manfred G. Madritsch

Modelling the LLL algorithm by sandpiles

The LLL algorithm aims at finding a “reduced” basis of a Euclidean lattice and plays a primary role in many areas of mathematics and computer science. However, its general behaviour is far from being well understood. There are already many experimental observations about the number of iterations or the geometry of the output, that raise challenging questions which remain unanswered and lead to natural conjectures which are yet to be proved. However, until now, there exist few experimental observations about the precise execution of the algorithm. Here, we provide experimental results which precisely describe an essential parameter of the execution, namely the “logarithm of the decreasing ratio”. These experiments give arguments towards a “regularity” hypothesis (R). Then, we propose a simplified model for the LLL algorithm based on the hypothesis (R), which leads us to discrete dynamical systems, namely sandpiles models. It is then possible to obtain a precise quantification of the main parameters of the LLL algorithm. These results fit the experimental results performed on general input bases, which indirectly substantiates the validity of such a regularity hypothesis and underlines the usefulness of such a simplified model.

Forms of differing degrees over number fields

Consider a system of polynomials in many variables over the ring of integers of a number field

Construction of normal numbers via pseudo-polynomial prime sequences

K

A note on generalized radix representations and dynamical systems

. We prove an asymptotic formula for the number of integral zeros of this system in homogeneously expanding boxes. As a consequence, any smooth and geometrically integral variety

Dynamical systems and uniform distribution of sequences

X\subseteq \mathbb{P}_K^m

q-ADDITIVE FUNCTIONS ON POLYNOMIAL SEQUENCES

satisfies the Hasse principle, weak approximation and the Manin-Peyre conjecture, if only its dimension is large enough compared to its degree. This generalizes work of Skinner, who considered the case where all polynomials have the same degree, and recent work of Browning and Heath-Brown, who considered the case where

On Multiplicative Independent Bases for Canonical Number Systems in Cyclotomic Number Fields

K=\mathbb{Q}

Waring’s problem with digital restrictions in \( \mathbb{F} \)q[X]

. Our main tool is Skinners number field version of the Hardy-Littlewood circle method. As a by-product, we point out and correct an error in Skinners treatment of the singular integral.

Normality of numbers generated by the values of entire functions

In the present paper we construct normal numbers in base

Asymptotic normality of additive functions on polynomial sequences in canonical number systems

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