arXiv subjects

Thomas Kahle

Publications and source records attributed to Thomas Kahle.

At least 37 records · Page 2Linked to original sources

arXiv2019

Multistationarity in the space of total concentrations for systems that admit a monomial parametrization

We apply tools from real algebraic geometry to the problem of multistationarity of chemical reaction networks. A particular focus is on the case of reaction networks whose steady states admit a monomial parametrization. For such systems we show that in the space of total concentrations multistationarity is scale invariant: if there is multistationarity for some value of the total concentrations, then there is multistationarity on the entire ray containing this value (possibly for different rate constants) -- and vice versa. Moreover, for these networks it is possible to decide about multistationarity independent of the rate constants by formulating semi-algebraic conditions that involve only concentration variables. These conditions can easily be extended to include total concentrations. Hence quantifier elimination may give new insights into multistationarity regions in the space of total concentrations. To demonstrate this, we show that for the distributive phosphorylation of a protein at two binding sites multistationarity is only possible if the total concentration of the substrate is larger than either the total concentration of the kinase or the total concentration of the phosphatase. This result is enabled by the chamber decomposition of the space of total concentrations from polyhedral geometry. Together with the corresponding sufficiency result of Bihan et al. this yields a characterization of multistationarity up to lower dimensional regions.

q-bio.MN↗

arXiv2019

Linear syzygies, hyperbolic Coxeter groups and regularity

We show that the virtual cohomological dimension of a Coxeter group is essentially the regularity of the Stanley--Reisner ring of its nerve. Using this connection between geometric group theory and commutative algebra, as well as techniques from the theory of hyperbolic Coxeter groups, we study the behavior of the Castelnuovo--Mumford regularity of square-free quadratic monomial ideals. We construct examples of such ideals which exhibit arbitrarily high regularity after linear syzygies for arbitrarily many steps. We give a doubly logarithmic bound on the regularity as a function of the number of variables if these ideals are Cohen--Macaulay.

math.AC↗

arXiv2018

The Geometry of Gaussoids

A gaussoid is a combinatorial structure that encodes independence in probability and statistics, just like matroids encode independence in linear algebra. The gaussoid axioms of Lnenicka and Matús are equivalent to compatibility with certain quadratic relations among principal and almost-principal minors of a symmetric matrix. We develop the geometric theory of gaussoids, based on the Lagrangian Grassmannian and its symmetries. We introduce oriented gaussoids and valuated gaussoids, thus connecting to real and tropical geometry. We classify small realizable and non-realizable gaussoids. Positive gaussoids are as nice as positroids: they are all realizable via graphical models.

math.CO↗

arXiv2018

Obstructions to combinatorial formulas for plethysm

Motivated by questions of Mulmuley and Stanley we investigate quasi-polynomials arising in formulas for plethysm. We demonstrate, on the examples of $S^3(S^k)$ and $S^k(S^3)$, that these need not be counting functions of inhomogeneous polytopes of dimension equal to the degree of the quasi-polynomial. It follows that these functions are not, in general, counting functions of lattice points in any scaled convex bodies, even when restricted to single rays. Our results also apply to special rectangular Kronecker coefficients.

math.RT↗

arXiv2017

Algebraic Aspects of Conditional Independence and Graphical Models

This chapter of the forthcoming Handbook of Graphical Models contains an overview of basic theorems and techniques from algebraic geometry and how they can be applied to the study of conditional independence and graphical models. It also introduces binomial ideals and some ideas from real algebraic geometry. When random variables are discrete or Gaussian, tools from computational algebraic geometry can be used to understand implications between conditional independence statements. This is accomplished by computing primary decompositions of conditional independence ideals. As examples the chapter presents in detail the graphical model of a four cycle and the intersection axiom, a certain implication of conditional independence statements. Another important problem in the area is to determine all constraints on a graphical model, for example, equations determined by trek separation. The full set of equality constraints can be determined by computing the model's vanishing ideal. The chapter illustrates these techniques and ideas with examples from the literature and provides references for further reading.

math.ST↗

arXiv2017

Finding binomials in polynomial ideals

We describe an algorithm which finds binomials in a given ideal $I\subset\mathbb{Q}[x_1,\dots,x_n]$ and in particular decides whether binomials exist in $I$ at all. Binomials in polynomial ideals can be well hidden. For example, the lowest degree of a binomial cannot be bounded as a function of the number of indeterminates, the degree of the generators, or the Castelnuovo--Mumford regularity. We approach the detection problem by reduction to the Artinian case using tropical geometry. The Artinian case is solved with algorithms from computational number theory.

math.AC↗

arXiv2017

Parity binomial edge ideals

Parity binomial edge ideals of simple undirected graphs are introduced. Unlike binomial edge ideals, they do not have square-free Gröbner bases and are radical if only if the graph is bipartite or the characteristic of the ground field is not two. The minimal primes are determined and shown to encode combinatorics of even and odd walks in the graph. A mesoprimary decomposition is determined and shown to be a primary decomposition in characteristic two.

math.AC↗

arXiv2016

The geometry of rank-one tensor completion

The geometry of the set of restrictions of rank-one tensors to some of their coordinates is studied. This gives insight into the problem of rank-one completion of partial tensors. Particular emphasis is put on the semialgebraic nature of the problem, which arises for real tensors with constraints on the parameters. The algebraic boundary of the completable region is described for tensors parametrized by probability distributions and where the number of observed entries equals the number of parameters. If the observations are on the diagonal of a tensor of format $d\times\dots\times d$, the complete semialgebraic description of the completable region is found.

math.AG↗

arXiv2016

Veronesean almost binomial almost complete intersections

The second Veronese ideal $I_n$ contains a natural complete intersection $J_n$ generated by the principal $2$-minors of a symmetric $(n\times n)$-matrix. We determine subintersections of the primary decomposition of $J_n$ where one intersectand is omitted. If $I_n$ is omitted, the result is the other end of a complete intersection link as in liaison theory. These subintersections also yield interesting insights into binomial ideals and multigraded algebra. For example, if $n$ is even, $I_n$ is a Gorenstein ideal and the intersection of the remaining primary components of $J_n$ equals $J_n+\langle f \rangle$ for an explicit polynomial $f$ constructed from the fibers of the Veronese grading map.

math.AC↗

arXiv2016

On the feasibility of semi-algebraic sets in Poisson regression

Designing experiments for generalized linear models is difficult because optimal designs depend on unknown parameters. The local optimality approach is to study the regions in parameter space where a given design is optimal. In many situations these regions are semi-algebraic. We investigate regions of optimality using computer tools such as yalmip, qepcad, and Mathematica.

math.ST↗

arXiv2016

Eigenschemes and the Jordan canonical form

We study the eigenscheme of a matrix which encodes information about the eigenvectors and generalized eigenvectors of a square matrix. The two main results in this paper are this decomposition encodes the numeric data of the Jordan canonical form of the matrix. We also describe how the eigenscheme can be interpreted as the zero locus of a global section of the tangent bundle on projective space. This interpretation allows one to see eigenvectors and generalized eigenvectors of matrices from an alternative viewpoint.

math.AG↗

arXiv2016

Irreducible decomposition of binomial ideals

Building on coprincipal mesoprimary decomposition [Kahle and Miller, 2014], we combinatorially construct an irreducible decomposition of any given binomial ideal. In a parallel manner, for congruences in commutative monoids we construct decompositions that are direct combinatorial analogues of binomial irreducible decompositions, and for binomial ideals we construct decompositions into ideals that are as irreducible as possible while remaining binomial. We provide an example of a binomial ideal that is not an intersection of binomial irreducible ideals, thus answering a question of Eisenbud and Sturmfels [1996].

math.AC↗

arXiv2015

Algebraic geometry of Poisson regression

Designing experiments for generalized linear models is difficult because optimal designs depend on unknown parameters. Here we investigate local optimality. We propose to study for a given design its region of optimality in parameter space. Often these regions are semi-algebraic and feature interesting symmetries. We demonstrate this with the Rasch Poisson counts model. For any given interaction order between the explanatory variables we give a characterization of the regions of optimality of a special saturated design. This extends known results from the case of no interaction. We also give an algebraic and geometric perspective on optimality of experimental designs for the Rasch Poisson counts model using polyhedral and spectrahedral geometry.

math.ST↗

arXiv2015

Detecting Binomiality

Binomial ideals are special polynomial ideals with many algorithmically and theoretically nice properties. We discuss the problem of deciding if a given polynomial ideal is binomial. While the methods are general, our main motivation and source of examples is the simplification of steady state equations of chemical reaction networks. For homogeneous ideals we give an efficient, Gröbner-free algorithm for binomiality detection, based on linear algebra only. On inhomogeneous input the algorithm can only give a sufficient condition for binomiality. As a remedy we construct a heuristic toolbox that can lead to simplifications even if the given ideal is not binomial.

math.CO↗

arXiv2015

Plethysm and lattice point counting

We apply lattice point counting methods to compute the multiplicities in the plethysm of $GL(n)$. Our approach gives insight into the asymptotic growth of the plethysm and makes the problem amenable to computer algebra. We prove an old conjecture of Howe on the leading term of plethysm. For any partition $μ$ of 3,4, or 5 we obtain an explicit formula in $λ$ and $k$ for the multiplicity of $S^λ$ in $S^μ(S^k)$.

math.RT↗

arXiv2014

Linear syzygies, flag complexes, and regularity

We show that for every positive integer R there exist monomial ideals generated in degree two, with linear syzygies, and regularity of the quotient equal to R. Such examples can not be found among Gorenstein ideals since the regularity of their quotients is at most four. We also show that for most monomial ideals generated in degree two and with linear syzygies the regularity grows at most doubly logarithmically in the number of variables.

math.AC↗

arXiv2014

Toric fiber products versus Segre products

The toric fiber product is an operation that combines two ideals that are homogeneous with respect to a grading by an affine monoid. The Segre product is a related construction that combines two multigraded rings. The quotient ring by a toric fiber product of two ideals is a subring of the Segre product, but in general this inclusion is strict. We contrast the two constructions and show that any Segre product can be presented as a toric fiber product without changing the involved quotient rings. This allows to apply previous results about toric fiber products to the study of Segre products. We give criteria for the Segre product of two affine toric varieties to be dense in their toric fiber product, and for the map from the Segre product to the toric fiber product to be finite. We give an example that shows that the quotient ring of a toric fiber product of normal ideals need not be normal. In rings with Veronese type gradings, we find examples of toric fiber products that are always Segre products, and we show that iterated toric fiber products of Veronese ideals over Veronese rings are normal.

math.AC↗

arXiv2014

Decompositions of commutative monoid congruences and binomial ideals

Primary decomposition of commutative monoid congruences is insensitive to certain features of primary decomposition in commutative rings. These features are captured by the more refined theory of mesoprimary decomposition of congruences, introduced here complete with witnesses and associated prime objects. The combinatorial theory of mesoprimary decomposition lifts to arbitrary binomial ideals in monoid algebras. The resulting binomial mesoprimary decomposition is a new type of intersection decomposition for binomial ideals that enjoys computational efficiency and independence from ground field hypotheses. Binomial primary decompositions are easily recovered from mesoprimary decomposition.

math.AC↗