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Georg Regensburger

Publications and source records attributed to Georg Regensburger.

At least 19 recordsLinked to original sources
arXiv2026

Disguised complex balance via positive algebraic geometry

We study dynamical systems arising from reaction networks under mass-action kinetics. For certain choices of the rate constants (parameters), such systems are complex-balanced (vertex-balanced), which guarantees the existence of a unique positive equilibrium. Moreover, this equilibrium is asymptotically stable (admitting a global Lyapunov function) and linearly stable. In a series of recent papers, Craciun and collaborators introduced and studied disguised complex-balanced systems, that is, mass-action systems that are dynamically equal to auxiliary complex-balanced systems and therefore inherit their strong stability properties. Determining the parameter values for which a given system is disguised complex-balanced is a nontrivial algebraic problem. In this work, we show that the defining conditions for disguised complex-balanced equilibria naturally give rise to parametrized systems of polynomial inequalities. Using the framework for positive algebraic geometry developed by M\"uller and Regensburger, we reformulate these systems as binomial equations (on the disguised complex-balanced flux cone). Computing the disguised complex-balanced parameter locus can be viewed as a quantifier-elimination problem, and our approach eliminates the concentrations (state variables) from the problem. We illustrate our results using the running example of a recent paper by Boros et al.

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arXiv2025

The integro-differential closure of a commutative differential ring

An integro-differential ring is a differential ring that is closed under an integration operation satisfying the fundamental theorem of calculus. Via the Newton--Leibniz formula, a generalized evaluation is defined in terms of integration and differentiation. The induced evaluation is not necessarily multiplicative, which allows one to model functions with singularities and leads to generalized shuffle relations. In general, not every element of a differential ring has an antiderivative in the same ring. Starting from a commutative differential ring and a direct decomposition into integrable and non-integrable elements, we construct the free integro-differential ring. This integro-differential closure contains all nested integrals over elements of the original differential ring. Analyzing the structure of nested integrals independently of concrete integrands, we separate the algebraic structure from analytic properties arising from concrete functions. We exhibit the relations satisfied by generalized evaluations of products of nested integrals. Investigating these relations of constants, we characterize in terms of Lyndon words certain evaluations of products that determine all others. We also analyze the relation of the free integro-differential ring with the shuffle algebra. To preserve integrals in the original differential ring for computations in its integro-differential closure, we introduce the notion of quasi-integro-differential rings and give two adapted constructions of the integro-differential closure. Finally, in a given integro-differential ring, we consider the internal integro-differential closure of a differential subring and identify it as quotient of the free integro-differential ring by certain constants.

math.RA↗
arXiv2024

A SageMath Package for Elementary and Sign Vectors with Applications to Chemical Reaction Networks

We present our SageMath package elementary_vectors for computing elementary and sign vectors of real subspaces. In this setting, elementary vectors are support-minimal vectors that can be determined from maximal minors of a real matrix representing a subspace. By applying the sign function, we obtain the cocircuits of the corresponding oriented matroid, which in turn allow the computation of all sign vectors of a real subspace. As an application, we discuss sign vector conditions for existence and uniqueness of complex-balanced equilibria of chemical reaction networks with generalized mass-action kinetics. The conditions are formulated in terms of sign vectors of two subspaces arising from the stoichiometric coefficients and the kinetic orders of the reactions. We discuss how these conditions can be checked algorithmically, and we demonstrate the functionality of our package sign_vector_conditions in several examples.

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arXiv2024

Universal truth of operator statements via ideal membership

We introduce a framework for proving statements about linear operators by verification of ideal membership in a free algebra. More specifically, arbitrary first-order statements about identities of morphisms in preadditive semicategories can be treated. We present a semi-decision procedure for validity of such formulas based on computations with noncommutative polynomials. These algebraic computations automatically incorporate linearity and benefit from efficient ideal membership procedures. In the framework, domains and codomains of operators are modelled using many-sorted first-order logic. To eliminate quantifiers and function symbols from logical formulas, we apply Herbrand's theorem and Ackermann's reduction. The validity of the resulting formulas is shown to be equivalent to finitely many ideal memberships of noncommutative polynomials. We explain all relevant concepts and discuss computational aspects. Furthermore, we illustrate our framework by proving concrete operator statements assisted by our computer algebra software.

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arXiv2023

Sufficient conditions for linear stability of complex-balanced equilibria in generalized mass-action systems

Generalized mass-action systems are power-law dynamical systems arising from chemical reaction networks. Essentially, every nonnegative ODE model used in chemistry and biology (for example, in ecology and epidemiology) and even in economics and engineering can be written in this form. Previous results have focused on existence and uniqueness of special steady states (complex-balanced equilibria) for all rate constants, thereby ruling out multiple (special) steady states. Recently, necessary conditions for linear stability have been obtained. In this work, we provide sufficient conditions for the linear stability of complex-balanced equilibria for all rate constants (and also for the non-existence of other steady states). In particular, via sign-vector conditions (on the stoichiometric coefficients and kinetic orders), we guarantee that the Jacobian matrix is a $P$-matrix. Technically, we use a new decomposition of the graph Laplacian which allows to consider orders of (generalized) monomials. Alternatively, we use cycle decomposition which allows a linear parametrization of all Jacobian matrices. In any case, we guarantee stability without explicit computation of steady states. We illustrate our results in examples from chemistry and biology: generalized Lotka-Volterra systems and SIR models, a two-component signaling system, and an enzymatic futile cycle.

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arXiv2023

How to automatise proofs of operator statements: Moore-Penrose inverse -- a case study

We describe a recently developed algebraic framework for proving first-order statements about linear operators by computations with noncommutative polynomials. Furthermore, we present our new SageMath package operator_gb, which offers functionality for automatising such computations. We aim to provide a practical understanding of our approach and the software through examples, while also explaining the completeness of the method in the sense that it allows to find algebraic proofs for every true first-order operator statement. We illustrate the capability of the framework in combination with our software by a case study on statements about the Moore-Penrose inverse, including classical facts and recent results, presented in an online notebook.

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arXiv2023

Parametrized systems of generalized polynomial inequalitites via linear algebra and convex geometry

We provide fundamental results on positive solutions to parametrized systems of generalized polynomial $\textit{inequalities}$ (with real exponents and positive parameters), including generalized polynomial $\textit{equations}$. In doing so, we also offer a new perspective on fewnomials and (generalized) mass-action systems. We find that geometric objects, rather than matrices, determine generalized polynomial systems: a bounded set/"polytope" $P$ (arising from the coefficient matrix) and two subspaces representing monomial differences and dependencies (arising from the exponent matrix). The dimension of the latter subspace, the monomial dependency $d$, is crucial. As our main result, we rewrite $\textit{polynomial inequalities}$ in terms of $d$ $\textit{binomial equations}$ on $P$, involving $d$ monomials in the parameters. In particular, we establish an explicit bijection between the original solution set and the solution set on $P$ via exponentiation. (i) Our results apply to any generalized polynomial system. (ii) The dependency $d$ and the dimension of $P$ indicate the complexity of a system. (iii) Our results are based on methods from linear algebra and convex/polyhedral geometry, and the solution set on $P$ can be further studied using methods from analysis such as sign-characteristic functions (introduced in this work). We illustrate our results (in particular, the relevant geometric objects) through three examples from real fewnomial and reaction network theory. For two mass-action systems, we parametrize the set of equilibria and the region for multistationarity, respectively, and even for univariate trinomials, we offer new insights: We provide a "solution formula" involving discriminants and "roots".

math.AG↗
arXiv2023

Parametrized systems of generalized polynomial equations: first applications to fewnomials

We consider positive solutions to parametrized systems of generalized polynomial equations (with real exponents and positive parameters). By a fundamental result obtained in parallel work, polynomial systems are determined by geometric objects, rather than matrices: a polytope $P$ (arising from the coefficient matrix) and two subspaces representing monomial differences and dependencies (arising from the exponent matrix). The dimension of the latter subspace, the monomial dependency $d$, is crucial. Indeed, we rewrite $\textit{polynomial}$ equations in terms of $d$ $\textit{binomial}$ equations on the coefficient polytope $P$, involving $d$ monomials in the parameters. We further study the solution set on $P$ using methods from analysis such as sign-characteristic functions and Wronskians. In this work, we present first applications to fewnomial systems through five (classes of) examples. In particular, we study (i) $n$ trinomials involving ${n+2}$ monomials in $n$ variables, having dependency $d=1$, and (ii) one trinomial and one $t$-nomial (with $t\ge3$) in two variables, having $d=t-1\ge2$. For (i), we bound the number of positive solutions using the number of roots of a univariate polynomial of degree at most $n$. We also show that this number is always less than or equal to the number of sign changes in an optimal Descartes' rule given in Bihan et al. (2021). For (ii), we improve upper bounds given in Li et al. (2003) and Koiran et al. (2015). Further, for two trinomials ($t=3$), we refine the known upper bound of five in terms of the exponents, and we find an example with five positive solutions that is even simpler than the smallest "Haas system".

math.AG↗
arXiv2023

The fundamental theorem of calculus in differential rings

In this paper, we study the consequences of the fundamental theorem of calculus from an algebraic point of view. For functions with singularities, this leads to a generalized notion of evaluation. We investigate properties of such integro-differential rings and discuss many examples. We also construct corresponding integro-differential operators and provide normal forms via rewrite rules. They are then used to derive several identities and properties in a purely algebraic way, generalizing well-known results from analysis. In identities like shuffle relations for nested integrals and the Taylor formula, additional terms are obtained that take singularities into account. Another focus lies on treating basics of linear ODEs in this framework of integro-differential operators. These operators can have matrix coefficients, which allow to treat systems of arbitrary size in a unified way. In the appendix, using tensor reduction systems, we give the technical details of normal forms and prove them for operators including other functionals besides evaluation.

math.RA↗
arXiv2022

Computing elements of certain form in ideals to prove properties of operators

Proving statements about linear operators expressed in terms of identities often leads to finding elements of certain form in noncommutative polynomial ideals. We illustrate this by examples coming from actual operator statements and discuss relevant algorithmic methods for finding such polynomials based on noncommutative Gröbner bases. In particular, we present algorithms for computing the intersection of a two-sided ideal with a one-sided ideal as well as for computing homogeneous polynomials in two-sided ideals and monomials in one-sided ideals. All methods presented in this work are implemented in the Mathematica package OperatorGB.

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arXiv2020

Algebraic proof methods for identities of matrices and operators: improvements of Hartwig's triple reverse order law

When improving results about generalized inverses, the aim often is to do this in the most general setting possible by eliminating superfluous assumptions and by simplifying some of the conditions in statements. In this paper, we use Hartwig's well-known triple reverse order law as an example for showing how this can be done using a recent framework for algebraic proofs and the software package OperatorGB. Our improvements of Hartwig's result are proven in rings with involution and we discuss computer-assisted proofs that show these results in other settings based on the framework and a single computation with noncommutative polynomials.

math.FA↗
arXiv2020

Compatible rewriting of noncommutative polynomials for proving operator identities

The goal of this paper is to prove operator identities using equalities between noncommutative polynomials. In general, a polynomial expression is not valid in terms of operators, since it may not be compatible with domains and codomains of the corresponding operators. Recently, some of the authors introduced a framework based on labelled quivers to rigorously translate polynomial identities to operator identities. In the present paper, we extend and adapt the framework to the context of rewriting and polynomial reduction. We give a sufficient condition on the polynomials used for rewriting to ensure that standard polynomial reduction automatically respects domains and codomains of operators. Finally, we adapt the noncommutative Buchberger procedure to compute additional compatible polynomials for rewriting. In the package OperatorGB, we also provide an implementation of the concepts developed.

cs.SC↗
arXiv2019

Formal proofs of operator identities by a single formal computation

A formal computation proving a new operator identity from known ones is, in principle, restricted by domains and codomains of linear operators involved, since not any two operators can be added or composed. Algebraically, identities can be modelled by noncommutative polynomials and such a formal computation proves that the polynomial corresponding to the new identity lies in the ideal generated by the polynomials corresponding to the known identities. In order to prove an operator identity, however, just proving membership of the polynomial in the ideal is not enough, since the ring of noncommutative polynomials ignores domains and codomains. We show that it suffices to additionally verify compatibility of this polynomial and of the generators of the ideal with the labelled quiver that encodes which polynomials can be realized as linear operators. Then, for every consistent representation of such a quiver in a linear category, there exists a computation in the category that proves the corresponding instance of the identity. Moreover, by assigning the same label to several edges of the quiver, the algebraic framework developed allows to model different versions of an operator by the same indeterminate in the noncommutative polynomials.

math.RA↗
arXiv2019

Complex-balanced equilibria of generalized mass-action systems: Necessary conditions for linear stability

It is well known that, for mass-action systems, complex-balanced equilibria are asymptotically stable. For generalized mass-action systems, even if there exists a unique complex-balanced equilibrium (in every stoichiometric class and for all rate constants), it need not be stable. We first discuss several notions of matrix stability (on a linear subspace) such as D-stability and diagonal stability, and then we apply our abstract results to complex-balanced equilibria of generalized mass-action systems. In particular, we show that linear stability (on the stoichiometric subspace and for all rate constants) implies uniqueness. For cyclic networks, we characterize linear stability (in terms of D-stability of the Jacobian matrix); and for weakly reversible networks, we give necessary conditions for linear stability (in terms of D-semistability of the Jacobian matrices of all cycles in the network). Moreover, we show that, for classical mass-action systems, complex-balanced equilibria are not just asymptotically stable, but even diagonally stable (and hence linearly stable). Finally, we recall and extend characterizations of D-stability and diagonal stability for matrices of dimension up to three, and we illustrate our results by examples of irreversible cycles (of dimension up to three) and of reversible chains and S-systems (of arbitrary dimension).

math.DS↗
arXiv2019

On the bijectivity of families of exponential/generalized polynomial maps

We start from a parametrized system of $d$ generalized polynomial equations (with real exponents) for $d$ positive variables, involving $n$ generalized monomials with $n$ positive parameters. Existence and uniqueness of a solution for all parameters and for all right-hand sides is equivalent to the bijectivity of (every element of) a family of generalized polynomial/exponential maps. We characterize the bijectivity of the family of exponential maps in terms of two linear subspaces arising from the coefficient and exponent matrices, respectively. In particular, we obtain conditions in terms of sign vectors of the two subspaces and a nondegeneracy condition involving the exponent subspace itself. Thereby, all criteria can be checked effectively. Moreover, we characterize when the existence of a unique solution is robust with respect to small perturbations of the exponents or/and the coefficients. In particular, we obtain conditions in terms of sign vectors of the linear subspaces or, alternatively, in terms of maximal minors of the coefficient and exponent matrices. Finally, we present applications to chemical reaction networks with (generalized) mass-action kinetics.

math.AG↗
arXiv2019

Characterizing injectivity of classes of maps via classes of matrices

We present a framework for characterizing injectivity of classes of maps (on cosets of a linear subspace) by injectivity of classes of matrices. Using our formalism, we characterize injectivity of several classes of maps, including generalized monomial and monotonic (not necessarily continuous) maps. In fact, monotonic maps are special cases of {\em component-wise affine} maps. Further, we study compositions of maps with a matrix and other composed maps, in particular, rational functions. Our framework covers classical injectivity criteria based on mean value theorems for vector-valued maps and recent results obtained in the study of chemical reaction networks.

math.AG↗
arXiv2018

Planar S-systems: Global stability and the center problem

S-systems are simple examples of power-law dynamical systems (polynomial systems with real exponents). For planar S-systems, we study global stability of the unique positive equilibrium and solve the center problem. Further, we construct a planar S-system with two limit cycles.

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