Related papers: The predictable degree property, column reducednes…
We give a conditional proof of the Uniform Boundedness Conjecture of Morton and Silverman in the case of polynomials over number fields, assuming a standard conjecture in arithmetic geometry. Our technique simultaneously yields a dynamical…
The dimensions of sets of matrices of various types, with specified eigenvalue multiplicities, are determined. The dimensions of the sets of matrices with given Jordan form and with given singular value multiplicities are also found. Each…
In this thesis new objects to the existing set of invariants of Lie algebras are added. These invariant characteristics are capable of describing the nilpotent parametric continuum of Lie algebras. The properties of these invariants, in…
We study the properties of the Minimum Description Length principle for sequence prediction, considering a two-part MDL estimator which is chosen from a countable class of models. This applies in particular to the important case of…
A relatively new topic in computability theory is the study of notions of computation that are robust against mistakes on some kind of small set. However, despite the recent popularity of this topic relatively foundational questions about…
Multi-dimensional classification (MDC) can be employed in a range of applications where one needs to predict multiple class variables for each given instance. Many existing MDC methods suffer from at least one of inaccuracy, scalability,…
Arthur Cohn's irreducibility criterion for polynomials with integer coefficients and its generalization connect primes to irreducibles, and integral bases to the variable $x$. As we follow this link, we find that these polynomials are ready…
The theory of algebras with polynomial identities has developed significantly, with special attention devoted to the classification of varieties according to the asymptotic behavior of their codimension sequences. This sequence is a…
The use of high-dimensional features has become a normal practice in many computer vision applications. The large dimension of these features is a limiting factor upon the number of data points which may be effectively stored and processed,…
We produce minimal integrity bases for both isotropic and hemitropic invariant algebras (and more generally covariant algebras) of most common bidimensional constitutive tensors and -- possibly coupled -- laws, including piezoelectricity…
Observable convolutional codes defined over Zpr with the Predictable Degree Property admit minimal input/state/output representations that preserve structural properties under scalar restriction. We make use of this fact to present…
We answer the following long-standing question of Kolchin: given a system of algebraic-differential equations $\Sigma(x_1,\dots,x_n)=0$ in $m$ derivatives over a differential field of characteristic zero, is there a computable bound, that…
Given a zero-dimensional ideal I in a polynomial ring, many computations start by finding univariate polynomials in I. Searching for a univariate polynomial in I is a particular case of considering the minimal polynomial of an element in…
Basic algebraic and combinatorial properties of finite vector spaces in which individual vectors are allowed to have multiplicities larger than $ 1 $ are derived. An application in coding theory is illustrated by showing that multispace…
Classification is a central problem for dynamical systems, in particular for families that arise in a wide range of topics, like substitution subshifts. It is important to be able to distinguish whether two such subshifts are isomorphic,…
Detailed information about the weight distribution of a convolutional code is given by the adjacency matrix of the state diagram associated with a controller canonical form of the code. We will show that this matrix is an invariant of the…
Computable reducibility is a well-established notion that allows to compare the complexity of various equivalence relations over the natural numbers. We generalize computable reducibility by introducing degree spectra of reducibility and…
Let F = F_p for any fixed prime p >= 2. An affine-invariant property is a property of functions on F^n that is closed under taking affine transformations of the domain. We prove that all affine-invariant property having local…
We give a first-order definition of key polynomials, we show the links with previous definitions, that it is relevant to study key degrees, and to use a kind of valuations that we call partially multiplicative. We also prove or reprove…
The paper deals with the problem of deciding if two finite-dimensional linear subspaces over an arbitrary field are identical up to a permutation of the coordinates. This problem is referred to as the permutation code equivalence. We show…