Differential elimination by differential specialization of Sylvester style matrices

Differential resultant formulas are defined, for a system \cP of $n$ ordinary Laurent differential polynomials in $n-1$ differential variables. These are determinants of coefficient matrices of an extended system of polynomials obtained from \cP through derivations and multiplications by Laurent monomials. To start, through derivations, a system \ps(\cP) of $L$ polynomials in $L-1$ algebraic variables is obtained, which is non sparse in the order of derivation. This enables the use of existing formulas for the computation of algebraic resultants, of the multivariate sparse algebraic polynomials in \ps(\cP), to obtain polynomials in the differential elimination ideal generated by \cP. The formulas obtained are multiples of the sparse differential resultant defined by Li, Yuan and Gao, and provide order and degree bounds in terms of mixed volumes in the generic case

Toric Sylvester forms and applications in elimination theory

In this paper, we investigate the structure of the saturation of ideals generated by square systems of sparse homogeneous polynomials over a toric variety $X$ with respect to the irrelevant ideal of $X$. As our main results, we establish a duality property and make it explicit by introducing toric Sylvester forms, under a certain positivity assumption on $X$. In particular, we prove that toric Sylvester forms yield bases of some graded components of $I^{sat}/I$, where $I$ denotes an ideal generated by $n + 1$ generic forms, $n$ is the dimension of $X$ and $I$ sat the saturation of $I$ with respect to the irrelevant ideal of the Cox ring of $X$. Then, to illustrate the relevance of toric Sylvester forms we provide three consequences in elimination theory: (1) we introduce a new family of elimination matrices that can be used to solve sparse polynomial systems by means of linear algebra methods, including overdetermined polynomial systems; (2) by incorporating toric Sylvester forms to the classical Koszul complex associated to a polynomial system, we obtain new expressions of the sparse resultant as a determinant of a complex; (3) we prove a new formula for computing toric residues of the product of two forms.Comment: 25 pages, 1 figur

Algorithmic Contributions to the Theory of Regular Chains

Regular chains, introduced about twenty years ago, have emerged as one of the major tools for solving polynomial systems symbolically. In this thesis, we focus on different algorithmic aspects of the theory of regular chains, from theoretical questions to high- performance implementation issues. The inclusion test for saturated ideals is a fundamental problem in this theory. By studying the primitivity of regular chains, we show that a regular chain generates its saturated ideal if and only if it is primitive. As a result, a family of inclusion tests can be detected very efficiently. The algorithm to compute the regular GCDs of two polynomials modulo a regular chain is one of the key routines in the various triangular decomposition algorithms. By revisiting relations between subresultants and GCDs, we proposed a novel bottom-up algorithm for this task, which improves the previous algorithm in a significant manner and creates opportunities for parallel execution. This thesis also discusses the accelerations towards fast Fourier transform (FFT) over finite fields and FFT based subresultant chain constructions in the context of massively parallel GPU architectures, which speedup our algorithms by several orders of magnitude

Analytic combinatorics : functional equations, rational and algebraic functions

This report is part of a series whose aim is to present in a synthetic way the major methods and models in analytic combinatorics. Here, we detail the case of rational and algebraic functions and discuss systematically closure properties, the location of singularities, and consequences regarding combinatorial enumeration. The theory is applied to regular and context-free languages, finite state models, paths in graphs, locally constrained permutati- ons, lattice paths and walks, trees, and planar maps

Information Geometry

This Special Issue of the journal Entropy, titled “Information Geometry I”, contains a collection of 17 papers concerning the foundations and applications of information geometry. Based on a geometrical interpretation of probability, information geometry has become a rich mathematical field employing the methods of differential geometry. It has numerous applications to data science, physics, and neuroscience. Presenting original research, yet written in an accessible, tutorial style, this collection of papers will be useful for scientists who are new to the field, while providing an excellent reference for the more experienced researcher. Several papers are written by authorities in the field, and topics cover the foundations of information geometry, as well as applications to statistics, Bayesian inference, machine learning, complex systems, physics, and neuroscience