15 research outputs found
Bounds for degrees of syzygies of polynomials defining a grade two ideal
We make explicit the exponential bound on the degrees of the polynomials appearing in the Effective Quillen-Suslin Theorem, and apply it jointly with the Hilbert-Burch Theorem to show that the syzygy module of a sequence of polynomials in variables defining a complete intersection ideal of grade two is free, and that a basis of it can be computed with bounded degrees. In the known cases, these bounds improve previous results
Counting Solutions of a Polynomial System Locally and Exactly
We propose a symbolic-numeric algorithm to count the number of solutions of a
polynomial system within a local region. More specifically, given a
zero-dimensional system , with
, and a polydisc
, our method aims to certify the existence
of solutions (counted with multiplicity) within the polydisc.
In case of success, it yields the correct result under guarantee. Otherwise,
no information is given. However, we show that our algorithm always succeeds if
is sufficiently small and well-isolating for a -fold
solution of the system.
Our analysis of the algorithm further yields a bound on the size of the
polydisc for which our algorithm succeeds under guarantee. This bound depends
on local parameters such as the size and multiplicity of as well
as the distances between and all other solutions. Efficiency of
our method stems from the fact that we reduce the problem of counting the roots
in of the original system to the problem of solving a
truncated system of degree . In particular, if the multiplicity of
is small compared to the total degrees of the polynomials ,
our method considerably improves upon known complete and certified methods.
For the special case of a bivariate system, we report on an implementation of
our algorithm, and show experimentally that our algorithm leads to a
significant improvement, when integrated as inclusion predicate into an
elimination method
Analytic Combinatorics in Several Variables: Effective Asymptotics and Lattice Path Enumeration
The field of analytic combinatorics, which studies the asymptotic behaviour
of sequences through analytic properties of their generating functions, has led
to the development of deep and powerful tools with applications across
mathematics and the natural sciences. In addition to the now classical
univariate theory, recent work in the study of analytic combinatorics in
several variables (ACSV) has shown how to derive asymptotics for the
coefficients of certain D-finite functions represented by diagonals of
multivariate rational functions. We give a pedagogical introduction to the
methods of ACSV from a computer algebra viewpoint, developing rigorous
algorithms and giving the first complexity results in this area under
conditions which are broadly satisfied. Furthermore, we give several new
applications of ACSV to the enumeration of lattice walks restricted to certain
regions. In addition to proving several open conjectures on the asymptotics of
such walks, a detailed study of lattice walk models with weighted steps is
undertaken.Comment: PhD thesis, University of Waterloo and ENS Lyon - 259 page
Valuative lattices and spectra
The first part of the present article consists in a survey about the
dynamical constructive method designed using dynamical theories and dynamical
algebraic structures. Dynamical methods uncovers a hidden computational content
for numerous abstract objects of classical mathematics, which seem a priori
inaccessible constructively, e.g., the algebraic closure of a (discrete) field.
When a proof in classical mathematics uses these abstract objects and results
in a concrete outcome, dynamical methods generally make possible to discover an
algorithm for this concrete outcome. The second part of the article applies
this dynamical method to the theory of divisibility. We compare two notions of
valuative spectra present in the literature and we introduce a third notion,
which is implicit in an article devoted to the dynamical theory of
algebraically closed discrete valued fields. The two first notions are
respectively due to Huber \& Knebusch and to Coquand. We prove that the
corresponding valuative lattices are essentially the same. We establish formal
Valuativestellens\"atze corresponding to these theories, and we compare the
various resulting notions of valuative dimensions.Comment: This file contains also a French version of the paper. English
version appears in the Proceedings of Graz Conference on Rings and
Factorizations 2021. Title: Algebraic, Number Theoretic, and Topological
Aspects of Ring Theory. Editors: Jean-Luc Chabert, Marco Fontana, Sophie
Frisch, Sarah Glaz, Keith Johnson. Springer 2023 ISBN 978-3-031-28846-3 DOI
10.1007/978-3-031-28847-
Paul Lorenzen -- Mathematician and Logician
This open access book examines the many contributions of Paul Lorenzen, an outstanding philosopher from the latter half of the 20th century. It features papers focused on integrating Lorenzen's original approach into the history of logic and mathematics. The papers also explore how practitioners can implement Lorenzen’s systematical ideas in today’s debates on proof-theoretic semantics, databank management, and stochastics. Coverage details key contributions of Lorenzen to constructive mathematics, Lorenzen’s work on lattice-groups and divisibility theory, and modern set theory and Lorenzen’s critique of actual infinity. The contributors also look at the main problem of Grundlagenforschung and Lorenzen’s consistency proof and Hilbert’s larger program. In addition, the papers offer a constructive examination of a Russell-style Ramified Type Theory and a way out of the circularity puzzle within the operative justification of logic and mathematics. Paul Lorenzen's name is associated with the Erlangen School of Methodical Constructivism, of which the approach in linguistic philosophy and philosophy of science determined philosophical discussions especially in Germany in the 1960s and 1970s. This volume features 10 papers from a meeting that took place at the University of Konstanz