Learning algebraic decompositions using Prony structures

Some algebraic aspects of mesoprimary decomposition

Recent results of Kahle and Miller give a method of constructing primary decompositions of binomial ideals by first constructing "mesoprimary decompositions" determined by their underlying monoid congruences. Monoid congruences (and therefore, binomial ideals) can present many subtle behaviors that must be carefully accounted for in order to produce general results, and this makes the theory complicated. In this paper, we examine their results in the presence of a positive $A$-grading, where certain pathologies are avoided and the theory becomes more accessible. Our approach is algebraic: whil…

Strongly étale difference algebras and Babbitt's decomposition

We introduce a class of strongly étale difference algebras, whose role in the study of difference equations is analogous to the role of étale algebras in the study of algebraic equations. We deduce an improved version of Babbitt's decomposition theorem and we present applications to difference algebraic groups and the compatibility problem.

Complexity of Triangular Representations of Algebraic Sets

Triangular decomposition is one of the standard ways to represent the radical of a polynomial ideal. A general algorithm for computing such a decomposition was proposed by A. Szanto. In this paper, we give the first complete bounds for the degrees of the polynomials and the number of components in the output of the algorithm, providing explicit formulas for these bounds.

Algebraic Geometry tools for the study of entanglement: an application to spin squeezed states

A short review of Algebraic Geometry tools for the decomposition of tensors and polynomials is given from the point of view of applications to quantum and atomic physics. Examples of application to assemblies of indistinguishable two-level bosonic atoms are discussed using modern formulations of the classical Sylvester's algorithm for the decomposition of homogeneous polynomials in two variables. In particular, the symmetric rank and symmetric border rank of spin squeezed states is calculated as well as their Schrödinger-cat-like decomposition as the sum of macroscopically different coherent s…

On product decomposition

Given a finite set $W$ in $\bar{k}^n$ where $\bar{k}$ is the algebraic closure of a field $k$ one would like to determine if $W$ can be decomposed as $\prod_{i=1}^n V_i$ where $V_i \subset \bar{k}$ under a linear transformation, that is, $W\stackrelλ{\to} \prod_{i=1}^n V_i$ where $λ\in Gl_n (\bar{k})$. We assume that $W$ is presented as $W=Z(\mathcal{F})$, the zero set of a polynomial system $\mathcal{F}$ in $n$ variables over $k$. We study algebraic characterization of such product decomposition. For decomposition into component sets of the same cardinality we obtain a stronger characterizati…

一种用于等维分解的Direttissimo算法

A Direttissimo Algorithm for Equidimensional Decomposition

本研究提出一种用于代数集等维分解的递归Direttissimo算法，结合三角集理论与Gröbner基处理局部闭代数集，无需投影及一般性假设，在结构化系统上可生成精细分解，实验验证其效率优于现有最优实现。