532 research outputs found
Explicit polynomial bounds on prime ideals in polynomial rings over fields
Suppose is an ideal of a polynomial ring over a field, , and whenever with degree , then either
or . When is sufficiently large, it follows that is
prime. Schmidt-G\"ottsch proved that "sufficiently large" can be taken to be a
polynomial in the degree of generators of (with the degree of this
polynomial depending on ). However Schmidt-G\"ottsch used model-theoretic
methods to show this, and did not give any indication of how large the degree
of this polynomial is. In this paper we obtain an explicit bound on ,
polynomial in the degree of the generators of . We also give a similar bound
for detecting maximal ideals in
Algorithmic Linearly Constrained Gaussian Processes
We algorithmically construct multi-output Gaussian process priors which
satisfy linear differential equations. Our approach attempts to parametrize all
solutions of the equations using Gr\"obner bases. If successful, a push forward
Gaussian process along the paramerization is the desired prior. We consider
several examples from physics, geomathematics and control, among them the full
inhomogeneous system of Maxwell's equations. By bringing together stochastic
learning and computer algebra in a novel way, we combine noisy observations
with precise algebraic computations.Comment: NIPS 201
Homotopy invariants for via Koszul duality
We show that the integral cohomology rings of the moduli spaces of stable
rational marked curves are Koszul. This answers an open question of Manin.
Using the machinery of Koszul spaces developed by Berglund, we compute the
rational homotopy Lie algebras of those spaces, and obtain some estimates for
Betti numbers of their free loop spaces in case of torsion coefficients. We
also prove and conjecture some generalisations of our main result.Comment: 14 pages, updated versio
Classification of regular parametrized one-relation operads
Jean-Louis Loday introduced a class of symmetric operads generated by one
bilinear operation subject to one relation making each left-normed product of
three elements equal to a linear combination of right-normed products:
such an operad is called a parametrized one-relation operad. For a
particular choice of parameters , this operad is said to be
regular if each of its components is the regular representation of the
symmetric group; equivalently, the corresponding free algebra on a vector space
is, as a graded vector space, isomorphic to the tensor algebra of . We
classify, over an algebraically closed field of characteristic zero, all
regular parametrized one-relation operads. In fact, we prove that each such
operad is isomorphic to one of the following five operads: the left-nilpotent
operad defined by the identity , the associative operad, the
Leibniz operad, the dual Leibniz (Zinbiel) operad, and the Poisson operad. Our
computational methods combine linear algebra over polynomial rings,
representation theory of the symmetric group, and Gr\"obner bases for
determinantal ideals and their radicals.Comment: 31 pages, final version, accepted for publication in Canadian Journal
of Mathematic
Involutive Algorithms for Computing Groebner Bases
In this paper we describe an efficient involutive algorithm for constructing
Groebner bases of polynomial ideals. The algorithm is based on the concept of
involutive monomial division which restricts the conventional division in a
certain way. In the presented algorithm a reduced Groebner basis is the
internally fixed subset of an involutive basis, and having computed the later,
the former can be output without any extra computational costs. We also discuss
some accounts of experimental superiority of the involutive algorithm over
Buchberger's algorithm.Comment: 27 pages, Proceedings of the NATO Advanced Research Workshop
"Computational commutative and non-commutative algebraic geometry"
(Chishinau, June 6-11, 2004), IOS Press, to appea
The degree of a tropical basis
We give an explicit upper bound for the degree of a tropical basis of a
homogeneous polynomial ideal. As an application f-vectors of tropical varieties
are discussed. Various examples illustrate differences between Gr\"obner and
tropical bases.Comment: 8 page
From analytical mechanical problems to rewriting theory through M. Janet
This note surveys the historical background of the Gr\"obner basis theory for
D-modules and linear rewriting theory. The objective is to present a deep
interaction of these two fields largely developed in algebra throughout the
twentieth century. We recall the work of M. Janet on the algebraic analysis on
linear partial differential systems that leads to the notion of involutive
division. We present some generalizations of the division introduced by M.
Janet and their relations with Gr\"obner basis theory.Comment: Lecture note of the Kobe-Lyon summer school 201
Ideals modulo a prime
The main focus of this paper is on the problem of relating an ideal in
the polynomial ring to a corresponding ideal in
where is a prime number; in other words, the
\textit{reduction modulo } of . We first define a new notion of
-good prime for which does depends on the term ordering ,
but not on the given generators of . We relate our notion of -good
primes to some other similar notions already in the literature.
Then we introduce and describe a new invariant called the universal
denominator which frees our definition of reduction modulo~ from the term
ordering, thus letting us show that all but finitely many primes are good for
.
One characteristic of our approach is that it enables us to easily detect
some bad primes, a distinct advantage when using modular methods.Comment: Improvements, and extended bibliography. To be published on "Journal
of Algebra and Its Applications (JAA)
Linear systems over localizations of rings
We describe a method for solving linear systems over the localization of a
commutative ring at a multiplicatively closed subset that works under
the following hypotheses: the ring is coherent, i.e., we can compute finite
generating sets of row syzygies of matrices over , and there is an algorithm
that decides for any given finitely generated ideal the
existence of an element in and in the affirmative case computes
as a concrete linear combination of the generators of .Comment: Improvement of the metho
Parameterized Type Definitions in Mathematica: Methods and Advantages
The theme of symbolic computation in algebraic categories has become of
utmost importance in the last decade since it enables the automatic modeling of
modern algebra theories. On this theoretical background, the present paper
reveals the utility of the parameterized categorical approach by deriving a
multivariate polynomial category (over various coefficient domains), which is
used by our Mathematica implementation of Buchberger's algorithms for
determining the Groebner basis. These implementations are designed according to
domain and category parameterization principles underlining their advantages:
operation protection, inheritance, generality, easy extendibility. In
particular, such an extension of Mathematica, a widely used symbolic
computation system, with a new type system has a certain practical importance.
The approach we propose for Mathematica is inspired from D. Gruntz and M.
Monagan's work in Gauss, for Maple.Comment: 14 page
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