The concept of invariant sets is an important tool to investigate the behaviour of solutions of ODE systems. A set $M$ is called invariant for a specified system if every trajectory of the system that starts in $M$ stays in $M$ for all times. Considering varieties and polynomial ODE systems, we show how Groebner bases provide an algorithmic test for invariance. As an application, we...
The strong Birch--Swinnerton-Dyer conjecture and in particular the exact order of the Shafarevich--Tate group for abelian varieties over the rationals has only been known for elliptic curves (dimension 1) or in higher dimension where the conjecture could be reduced to dimension 1. We give the first absolutely simple examples of dimension 2 where the conjecture can be verified:
Let $X$ be...
We give an introduction to the OSCAR Computer Algebra System (see https://oscar.computeralgebra.de) for computations in algebra, geometry, and number theory.
In this talk, we'll demonstrate how to implement categories on a computer using our software project CAP - Categories, Algorithms, and Programming. We will discuss how to construct the bounded homotopy category of an additive or Abelian category, as well as the obstacles that arise and their practical categorical solutions.
We use these homotopy categories to demonstrate some applications...
It is important that results published in papers are understandable and verifiable by other mathematicians. Even if they read them a hundred years from now. If a paper is highly dependent on a software component this may often be difficult. The software could be out of date, very difficult to install or contain bugs that were never checked for. Computed results might not be available anymore...
As can be seen in the preceding talk ("Constructive Category Theory and Tilting equivalences via Strong Exceptional Sequences" by Kamal Saleh), building constructive towers of categories allows us to reach advanced and complex applications while retaining (relative) simplicity of both the mathematical descriptions and the organization of the software. Our software project CAP (Categories,...
Given a first order autonomous algebraic ordinary differential equation, i.e. an equation of the form
F(y, y')=0 with F ∈ K[y, y'],
where K is an computable field such as the rational numbers, we present a method to compute all formal Puiseux series solutions. In fact, all of the solutions are convergent Puiseux series.
By considering y and y' as independent variables, F implicitly defines...
Wir zeigen die wichtigsten Neuerungen in Maple 2022 aus dem Bereich symbolische Mathematik, darüber hinaus einiges aus den Themen Numerik, Programmierung und Performance, Benutzeroberfläche und Grafik sowie Connectivity.
In group theory, we have a certain type of groups, i.e. algebraic groups. Thanks to an additional underlying structure, we have certain rules for the action of the groups on themselves and their associated Lie algebras.
We are interested in solving several problems concerning algebraic groups (for example linked to representation theory) in small characteristic. These complex problems often...
Power series remain the best tool for effective evaluations of combinations of elementary functions. We overview a recent result concerning the symbolic computation of univariate formal power series.
Given a field K, of characteristic zero, a term a(n) is called m-fold hypergeometric for a positive integer m (mostly m>1), if the ratio a(n+m)/a(n) is rational in K(n). When the value of m is...
We present a novel, practical method to determine the Frattini subgroup of a polycyclic group. This method is based on new theoretical investigations about complements and module strucure of elementary abelian sections in polycyclic groups. We have implemented our method in GAP and include a discussion of this implementation.
We introduce the integral matrix similarity problem as a special instance of the conjugacy problem from group theory. While this problem has been known to be decidable for over 40 years due to work of Grunewald and Sarkisyan, no upper bound on the complexity has been known and practical algorithms have only been available for special cases. We will present recent work with Werner Bley and...
We develop an algorithm based on a greedy algorithm of Cuntz from his 2021 paper "A Greedy Algorithm to Compute Arrangements of Lines in the Projective Plane," to find arrangements of lines for which a "Mod 2-net coloring" is possible. In such a coloring we color the lines of the arrangement with one of two colors so, that for all lines going through a given point, we have an even number of...
Homomorphisms of modules are fundamental objects in algebra and applied mathematics. Recent developments in symbolic computation allow a formal verification and exploration of identities between homomorphisms. Systems of those identities define varieties similar as in classical algebraic geometry. For constant-free systems one has a Galois connection to free algebras, and morphisms of...
Given a 0-dimensional projective scheme X in the projective n-space over a field K, we are interested in studying the Cayley-Bacharach property (CB-property) of X. Classically, a finite set of points X has the CB-property if every hypersurface of certain degree which contains all points of X but one automatically contains the last point. In this talk we shows how to check the CB-property of a...
Conditional independence is a ternary relation from probability theory between subcollections of jointly distributed random variables. Among normally distributed (i.e., "Gaussian") variables, these relations are characterized by the vanishing of specific subdeterminants of the distribution's positive definite covariance matrix.
The combinatorial structures realizable as conditional...
In this talk we shed light on varieties of moments arising from graphical models. A directed graph corresponds to a statistical model where the nodes represent random variables and arrows encode relations between them. The polynomial relations between entries of the covariance matrices have been previously studied from an algebraic and combinatorial point of view. Dropping Gaussianity makes...
An infinite sequence $\mathcal{ F } = (F_0, F_1, \dots)$ of function fields $F_n / \mathbb{F}_q $ of transcendence degree one with full constant field $ \mathbb{F}_q $ is called a tower of function fields if all extensions $F_{n + 1}/F_n$ are finite separable and the genus $g(F_n)$ tends to infinity as $n \to \infty$.
A tower is called asymptotically good if its limit $\lambda(\mathcal{F})...
Countless applied problems in various sciences can be expressed as polynomial optimization problems. Solving these nonlinear problems essentially requires to certify nonnegativity of multivariate, real polynomials, a classical problem from real algebraic geometry.
A classical way to certify nonnegativity are sums of squares (SOS). An alternative way are sums of nonnegative circuit polynomials...
It is very common nowadays to use tools from symbolic computation for applications in many different areas of mathematics (or physics, or computer science, etc.). One backbone of symbolic computation is Cylindrical Algebraic Decomposition (CAD). In the past years, we have applied CAD to problems arising in applied mathematics. Among the prominent methods in computer algebra are algorithms to...
