We present a SAGBI detection algorithm which, given a finite set of polynomials, determines all term orders for which this set forms a SAGBI basis. Based on this, we implement the SAGBI homotopy package in Julia using HomotopyContinuation.jl and Oscar.jl. The package provides a specific choice of start system to proceed with homotopy. For systems where each equation is a linear combination of fixed polynomials, SAGBI homotopies can significantly reduce the number of solution paths that need to be tracked compared to polyhedral homotopies. We will illustrate this approach on polynomial systems arising in applications from the sciences. This is based on joint works with Elima Shehu, Timothy Duff, and Barbara Betti.

I will present a semi-algorithm that takes an irreducible polynomial $p(x,y)$ and a rational function $r(x,y)$ and returns (a description of) all solutions of $r(x,y) + q(x,y) p(x,y) = f(x) - g(y)$ where $q(x,y)$ is a rational function whose denominator is not divisible by $p(x,y)$ and $f(x)$ and $g(y)$ are univariate rational functions. The semi-algorithm presented does not necessarily terminate. It does, if the equation has a (non-trivial) solution. However, if there is none, then it might not terminate. Termination depends on the curve defined by $p(x,y)$, the location of the poles of $r(x,y)$, and a dynamical system associated with them. This work originates from the study of discrete differential equations, a class of functional equations that arise in enumerative combinatorics, in the enumeration of restricted lattice walks. Further applications are: the computation of intersections of fields, which again has applications in computer vision, parameter identifiability of ODE models and the problem of showing algebraic (in)dependence of solutions of differential equations.

We discuss a recent approach to arbitrary systems of generalized polynomial inequalities in positive variables, where each monomial involves a distinct positive parameter. The key geometric objects in the problem are a polytope arising from the coefficients and two subspaces that capture differences and dependencies among the exponents. The dimension $d$ of the latter subspace, which we refer to as the monomial dependency, plays a central role. In our framework, based on linear algebra and convex geometry, we reformulate the problem in terms of $d$ binomial equations defined on the coefficient polytope, involving $d$ monomials in the positive parameters. We also discuss computational aspects and illustrate the method with applications to fewnomial theory. The talk is based on joint work with Stefan Müller

Given by an embedded graph equipped with Euclidean edge-length constraints, bar-joint frameworks provide a powerful model for various geometric constraint systems. As these constraints can be transformed into a system of quadratic polynomials, associated approaches from algebraic geometry such as Gröbner bases become applicable. Often, we are interesting in distinguishing between a framework’s rigidity and flexibility. In algebraic terms, this problem translates to deciding, whether a configuration is a real isolated solution of the underlying system of equations. Despite their simplicity, bar-joint frameworks find application in diverse real-world settings such as Computer Aided Design, Protein folding, polytope theory, sphere packings and formation control. In an effort to unify the existing algorithmic approaches in rigidity theory, my collaborators—Matteo Gallet, Georg Grasegger and Jan Legerský—and I developed PyRigi, an intuitive and user-friendly Python package. In this presentation, I will introduce basic results about bar-joint frameworks and provide an overview of the latest stable release of PyRigi. I will also introduce a new approach for efficiently and symbolically determining a framework’s second-order rigidity, as well as a homotopy continuation approach for numerically approximating deformation paths. Lastly, I will give an outlook on planned implementations and invite discussion on future directions beyond bar-joint rigidity.

We will give an insight into the computer algebra system OSCAR and how to use it. After a brief introduction to and background of the system, we will look into some easy-to-grasp real-life examples. OSCAR is being developed as part of the SFB-TRR 195 "Symbolic Tools in Mathematics and their Application". It is a general-purpose computer algebra system written in julia that builds on the four cornerstones GAP, polymake, Singular, and Antic (Hecke, Nemo) and has capabilities for dealing with problems in number theory, group and representation theory, tropical and polyhedral geometry, algebraic geometry, commutative algebra, non-commutative algebra, and many more areas of computer algebra.

Computer algebra has proven to be a powerful tool for the formal verification of circuits. In this approach, circuits are encoded as sets of polynomials forming a lexicographic Gröbner basis, and verification is performed by reducing the circuit’s specification to normal form. However, using the lexicographic Gröbner basis for this computation typically leads to a monomial blow-up of intermediate results. Recently, an alternative has emerged that shifts the computational effort from reducing the specification to rewriting the Gröbner basis itself. By a change of order, the lexicographic Gröbner basis is (partially) converted into a basis that includes more linear polynomials. Using these linear polynomials in the reduction of the specification mitigates the intermediate blow-up and simplifies the reduction process. Thus far, a black-box Gröbner basis computation has been used to rewrite the lexicographic basis, which has several drawbacks. In this talk, we present a different approach, in which we compute linear polynomials via a variant of the FGLM algorithm. This talk is based on joint work with Daniela Kaufmann (TU Vienna).

Given a representation $\rho:G \to \mathrm{GL}_n(K)$ for $G$ a finite group and $K \subseteq \mathbb{R}$, we can construct a non-degenerate symmetric bilinear form $\beta$ on $K^n$ which is $\rho(G)$-invariant, i.e., $\beta(\rho(g)v,\rho(g)w)=\beta(v,w)$ for any $g \in G$ and $v,w \in K^n$. If the square class $\det(\beta)=\det(\beta_0(e_i,e_j)_{1 \leq i,j \leq n}) \cdot (K^{\times})^2$ is independent of the chosen bilinear form, we say that $\rho$ is orthogonally stable and that $\det(\rho):=\det(\beta)$ is the orthogonal determinant of $\rho$. The notion extends to orthogonally stable characters and the orthogonal determinants theoreof. In this talk, we discuss methods and results to calculate the orthogonal determinants for finite groups of Lie type $G(q)$ (such as $\mathrm{GL}_n(q)$, $\mathrm{SU}_n(q)$, $F_4(q)$, $E_8(q)$, $\dots$) for $q$ a power of an odd prime. The calculations can basically be reduced to determining the orthogonal determinants of (irreducible) Iwahori--Hecke algebras, which consist of 4 infinite families and 6 exceptional cases. While theoretical results exist for the infinite families, explicit calculations have to be done for the exceptional cases, for which we use GAP and CHEVIE.

The aim of this talk is to review results obtained towards the calculation of trivial source character tables of "small" finite groups, and the creation of a database of such tables.

Based only on the Leibniz rule and the fundamental theorem of calculus, we introduced integro-differential rings with a generalized notion of evaluation that allows to treat functions with singularities. Such evaluations are not necessarily multiplicative, which leads to additional terms being introduced into well-known identities such as the Taylor-formula. In this talk, we discuss examples and basic properties of such integro-differential rings. Integro-differential operators involving this kind of evaluation can take more involved forms. We discuss normal forms and confluent rewrite rules for such operators that allow to derive new identities. Moreover, if a differential ring does not include an integral for each of its elements, we introduce the notion of quasi-integration. This is slightly stronger than additive decompositions, which split a function into an integrable and a non-integrable part and have been designed in symbolic integration for many different settings. Based on this notion, we present and discuss the free commutative integro-differential ring generated by a differential ring. This provides the algebraic structure for computing with nested integrals as soon as a quasi-integration is available in a given differential ring. In particular, starting from differential polynomials, this covers the construction of integro-differential polynomials. This is joint work with Georg Regensburger.

Let $G$ be one of the classical groups of Lie type with Lie rank $l$. Similar to the general polynomial equation for $S_n$ in classical Galois theory we constructed in differential Galois theory a general linear differential equation having differential Galois group $G$. More precisely, for an algebraically closed field $C$ of characteristic zero and $l$ differential indeterminates ${\bf v}=(v_1,\dots,v_l)$ let $C\langle {\bf v} \rangle$ be the differential field generated by ${\bf v}$ over $C$. Using the geometrical structure of $G$ we were able to construct a specific differential extension field $\mathcal{E}$ of $C\langle {\bf v} \rangle$ and to define an action of $G$ on $\mathcal{E}$ such that the fixed field under this action is differentially generated over $C$ by $l$ differential polynomials $${\bf s}({\bf v})=(s_1({\bf v}),\dots, s_l({\bf v})) \in C\{ {\bf v}\},$$ which take on the role of the elementary symmetric polynomials. Similarly as in the classical case, $\mathcal{E}$ is then a Picard-Vessiot extension of $C\langle {\bf s}({\bf v}) \rangle $ with differential Galois group $G$ and it is defined by a linear differential equation \[ L_G({\bf s}( {\bf v}),y) =0, \] whose coefficients depend on ${\bf s}({\bf v})$. We call $\mathcal{E}/C\langle {\bf s}({\bf v}) \rangle $ the {\it normal form extension} and $L_G({\bf s}({\bf v}),y) =0$ the {\it normal form equation} for $G$. In this talk we present an algorithm for the computation of the differential Galois group of a specialization \[ L_G(\bar{{\bf s}},y)=0 \] of our normal form equation with \[ \sigma_0 : C\{ {\bf s}({\bf v}) \} \rightarrow C(z), \quad {\bf s}({\bf v}) \mapsto \bar{{\bf s}} \in C(z)^l, \] where $C(z)$ denotes the rational function field in $z$ with standard derivation $\frac{d}{dz}$. Our algorithm is based on the combination of known algorithms for the computation of the differential Galois group and its Lie algebra in specific cases with our normal from extension.

A sequence is difference algebraic (or D-algebraic) if finitely many shifts of its general term satisfy a polynomial relationship; that is, they are the coordinates of a generic point on an affine hypersurface. For holonomic sequences (those defined by linear recurrence equations with polynomial coefficients), proving their D-algebraicity is equivalent to eliminating the index variable from their defining equations. What is special about their case is that they often satisfy a polynomial difference equation where the highest shift term appears linearly. We establish this result and discuss its implications. This is partly joint work with James Worrell.

A complete reduction on a difference field is a linear operator that enables one to decompose an element of the field as the sum of a summable part and a remainder such that the given element is summable if and only if the remainder is equal to zero. In this talk, we present a complete reduction in a tower of $\Sigma^*$-extensions that turns to a new algorithm for the parameterized telescoping problem. The underlying framework will be illustrated by concrete examples. This is joint work with Shaoshi Chen, Hui Huang and Carsten Schneider.