Related papers: Semidefinite Characterization and Computation of R…
This is a survey article on real algebra and geometry, and in particular on its recent applications in optimization and convexity. We first introduce basic notions and results from the classical theory. We then explain how these relate to…
Our focus in this paper is in effective computation of the core core(I) of an ideal I which is defined to be the intersection of all minimal reductions of I. The first main result is a closed formula for the graded core(m) of the maximal…
We present two quantum interior point methods for semidefinite optimization problems, building on recent advances in quantum linear system algorithms. The first scheme, more similar to a classical solution algorithm, computes an inexact…
Ideals generated by adjacent 2-minors are studied. First, the problem when such an ideal is a prime ideal as well as the problem when such an ideal possesses a quadratic Gr\"obner basis is solved. Second, we describe explicitly a primary…
The elements of a finite partial order $P$ can be identified with the maximal indecomposable two-sided ideals of its incidence algebra $\A$, and then for two such ideals, $I\prec J \iff IJ \not=0$. This offers one way to recover a poset…
Recent algorithmic advances in algebraic automata theory drew attention to semigroupoids (semicategories). These are mathematical descriptions of typed computational processes, but they have not been studied systematically in the context of…
First some old as well as new results about P.I. algebras, Ore extensions, and degrees are presented. Then quantized $n\times r$ matrices as well as quantized factor algebras of $M_q(n)$ are analyzed. The latter are the quantized function…
Reduced ideals have been defined in the context of integer rings in quadratic number fields, and they are closely tied to the continued fraction algorithm. The notion of this type of ideal extends naturally to number fields of higher…
In this paper, we present a modular strategy which describes key properties of the absolute primary decomposition of an equidimensional polynomial ideal defined by polynomials with rational coefficients. The algorithm we design is based on…
The main result provides an algorithm for determining the minimal free resolution of ideals of fat point subschemes of ${\bf P}^2$ involving up to 8 general points with arbitrary multiplicities; the results hold over algebraically closed…
Regular chains and triangular decompositions are fundamental and well-developed tools for describing the complex solutions of polynomial systems. This paper proposes adaptations of these tools focusing on solutions of the real analogue:…
We determine sets of elements which, under certain conditions, generate an intersection of ideals up to radical.
We will explore some properties of minimal graded free resolutions of $R/I$, where $R$ is a trivariate polynomial ring over a field and $I$ is a monomial ideal. Our focus will be to consider a specific form of the resolutions when $I$ is…
This paper provides an in-depth analysis of how computational algebraic geometry can be used to deal with the problem of counting and classifying $r\times s$ partial Latin rectangles based on $n$ symbols of a given size, shape, type or…
The numerical range of a matrix is studied geometrically via the cone of positive semidefinite matrices (or semidefinite cone for short). In particular it is shown that the feasible set of a two-dimensional linear matrix inequality (LMI),…
The numerical range of a matrix is studied geometrically via the cone of positive semidefinite matrices (or semidefinite cone for short). In particular it is shown that the feasible set of a two-dimensional linear matrix inequality (LMI),…
The accuracy and complexity of machine learning algorithms based on kernel optimization are determined by the set of kernels over which they are able to optimize. An ideal set of kernels should: admit a linear parameterization (for…
We state and give self contained proofs of semidefinite programming characterizations of the numerical radius and its dual norm for matrices. We show that the computation of the numerical radius and its dual norm within $\varepsilon$…
In this paper we present a new efficient variant to compute strong Gr\"obner basis over quotients of principal ideal domains. We show an easy lifting process which allows us to reduce one computation over the quotient $R/nR$ to two…
This paper presents generalizations of semidefinite programming formulations of 1-norm optimization problems over infinite dictionaries of vectors of complex exponentials, which were recently proposed for superresolution, gridless…