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In this paper we revisit the likelihood geometry of Gaussian graphical models. We give a detailed proof that the ML-degree behaves monotonically on induced subgraphs. Furthermore, we complete a missing argument that the ML-degree of the $n$-th cycle is larger than one for any $n\geq 4$, therefore completing the characterization that the only Gaussian graphical models with rational maximum likelihood estimator are the ones corresponding to chordal (decomposable) graphs. Finally, we prove that the formula for the ML-degree of a cycle conjectured by Drton, Sturmfels and Sullivant provides a correct lower bound.

We investigate border ranks of twisted powers of polynomials and smoothability of symmetric powers of algebras. We prove that the latter are smoothable. For the former, we obtain upper bounds for the border rank in general and prove that they are optimal under mild conditions. We give applications to complexity theory., Comment: 34 pages, comments welcome!

In topology, one averages over local geometrical details to reveal robust global features. This approach proves crucial for understanding quantized bulk transport and exotic boundary effects of linear wave propagation in (meta-)materials. Moving beyond linear Hamiltonian systems, the study of topology in physics strives to characterize open (non-Hermitian) and interacting systems. Here, we establish a framework for the topological classification of driven-dissipative nonlinear systems. Specifically, we define a graph index for the Floquet semiclassical equations of motion describing such systems. The graph index builds upon topological vector analysis theory and combines knowledge of the particle-hole nature of fluctuations around each out-of-equilibrium stationary state. To test this approach, we divulge the topological invariants arising in a micro-electromechanical nonlinear resonator subject to forcing and a time-modulated potential. Our framework classifies the complete phase diagram of the system and reveals the topological origin of driven-dissipative phase transitions, as well as that of under- to over-damped responses. Furthermore, we predict topological phase transitions between symmetry-broken phases that pertain to population inversion transitions. This rich manifesting phenomenology reveals the pervasive link between topology and nonlinear dynamics, with broad implications for all fields of science., Comment: 8 pages, 4 figures (G.V. and J.d.P. contributed equally to this work)

The exponent $\sigma(T)$ of a tensor $T\in\mathbb{F}^d\otimes\mathbb{F}^d\otimes\mathbb{F}^d$ over a field $\mathbb{F}$ captures the base of the exponential growth rate of the tensor rank of $T$ under Kronecker powers. Tensor exponents are fundamental from the standpoint of algorithms and computational complexity theory; for example, the exponent $\omega$ of matrix multiplication can be characterized as $\omega=2\sigma(\mathrm{MM}_2)$, where $\mathrm{MM}_2\in\mathbb{F}^4\otimes\mathbb{F}^4\otimes\mathbb{F}^4$ is the tensor that represents $2\times 2$ matrix multiplication. Our main result is an explicit construction of a sequence $\mathcal{U}_d$ of zero-one-valued tensors that is universal for the worst-case tensor exponent; more precisely, we show that $\sigma(\mathcal{U}_d)=\sigma(d)$ where $\sigma(d)=\sup_{T\in\mathbb{F}^d\otimes\mathbb{F}^d\otimes\mathbb{F}^d}\sigma(T)$. We also supply an explicit universal sequence $\mathcal{U}_\Delta$ localised to capture the worst-case exponent $\sigma(\Delta)$ of tensors with support contained in $\Delta\subseteq [d]\times[d]\times [d]$; by combining such sequences, we obtain a universal sequence $\mathcal{T}_d$ such that $\sigma(\mathcal{T}_d)=1$ holds if and only if Strassen's asymptotic rank conjecture [Progr. Math. 120 (1994)] holds for $d$. Finally, we show that the limit $\lim_{d\rightarrow\infty}\sigma(d)$ exists and can be captured as $\lim_{d\rightarrow\infty} \sigma(D_d)$ for an explicit sequence $(D_d)_{d=1}^\infty$ of tensors obtained by diagonalisation of the sequences $\mathcal{U}_d$. As our second result we relate the absence of polynomials of fixed degree vanishing on tensors of low rank, or more generally asymptotic rank, with upper bounds on the exponent $\sigma(d)$. Using this technique, one may bound asymptotic rank for all tensors of a given format, knowing enough specific tensors of low asymptotic rank.

In this article, we study permanental varieties, i.e. varieties defined by the vanishing of permanents of fixed size of a generic matrix. Permanents and their varieties play an important, and sometimes poorly understood, role in combinatorics. However, there are essentially no geometric results about them in the literature, in very sharp contrast to the well-behaved and ubiquitous case of determinants and minors. Motivated by the study of the singular locus of the permanental hypersurface, we focus on the codimension of these varieties. We introduce a $\mathbb C^{*}$-action on matrices and prove a number of results. In particular, we improve a lower bound on the codimension of the aforementioned singular locus established by von zur Gathen in 1987., Comment: 20p

In this paper, we settle a conjecture due to Sturmfels and Uhler concerning generation of the prime ideal of the variety associated to the Gaussian graphical model of any cycle graph. Our methods are general and applicable to a large class of ideals with radical initial ideals.

We provide an approach to counting roots of polynomial systems, where each polynomial is a general linear combination of prescribed, fixed polynomials. Our tools rely on the theory of Khovanskii bases, combined with toric geometry, the Bernstein-Khovanskii-Kushnirenko (BKK) Theorem, and fiber products. As a direct application of this theory, we solve the problem of counting the number of approximate stationary states for coupled driven nonlinear resonators. We set up a system of polynomial equations that depends on three numbers $N, n$ and $M$ and whose solutions model the stationary states. The parameter $N$ is the number of coupled resonators, $2n - 1$ is the degree of nonlinearity of the underlying differential equation, and $M$ is the number of frequencies used in the approximation. We use our main theorems, that is, the generalized BKK Theorem and the Decoupling Theorem, to count the number of (complex) solutions of the polynomial system for an arbitrary degree of nonlinearity $2n - 1 \geq 3$, any number of resonators $N \geq 1$, and $M = 1$ harmonic. We also solve the case $N = 1, n = 2$ and $M = 2$ and give a computational way to check the number of solutions for $N = 1, n = 2$ and $M \geq 2$. This extends the results of arXiv:2208.08179.

We provide a complete description of the ideal that serves as the resultant ideal for n univariate polynomials of degree d. We in particular describe a set of generators of this resultant ideal arising as maximal minors of a set of cascading matrices formed from the coefficients of the polynomials, generalising the classical Sylvester resultant of two polynomials., Comment: 9 pages; v2: added some classical references and a new result on projective scheme structure, communicated by Jan Stevens

By the work of J.Huh, one can interpret binomial coefficients as a solution to an intersection problem on a permutohedral variety $X_E$. Applying Hirzebruch-Riemann-Roch, this intersection problem is equivalent to computing Euler characteristic of a specific element of $K$-theory of $X_E$. This element has a natural lifting to equivariant $K$-theory and thus the Euler characteristic may be upgraded to a Laurent polynomial. We provide and implement three different approaches, in particular a recursive one, to computing these polynomials., Comment: 21 pages, 1 figure

We explain connections among several, a priori unrelated, areas of mathematics: combinatorics, algebraic statistics, topology and enumerative algebraic geometry. Our focus is on discrete invariants, strongly related to the theory of Lorentzian polynomials. The main concept joining the mentioned fields is a linear space of matrices.

Approximating periodic solutions to the coupled Duffing equations amounts to solving a system of polynomial equations. The number of complex solutions measures the algebraic complexity of this approximation problem. Using the theory of Khovanskii bases, we show that this number is given by the volume of a certain polytope. We also show how to compute all solutions using numerical nonlinear algebra.

We consider the inverse problem for the polynomial map which sends an $m$-tuple of quadratic forms in $n$ variables to the sum of their $d$-th powers. This map captures the moment problem for mixtures of $m$ centered $n$-variate Gaussians. In the first non-trivial case $d = 3$, we show that for any $ n\in \mathbb N $, this map is generically one-to-one (up to permutations of $ q_1,\ldots, q_m $ and third roots of unity) in two ranges: $m\le {n\choose 2} + 1 $ for $n \leq 16$ and $ m\le {n+5 \choose 6}/{n+1 \choose 2}-{n+1 \choose 2}-1$ for $n > 16$, thus proving generic identifiability for mixtures of centered Gaussians from their (exact) moments of degree at most $ 6 $. The first result is obtained by studying the explicit geometry of the tangential contact locus of the variety of sums of cubes of quadratic forms at concrete points, while the second result is accomplished using a link between secant non-defectivity with identifiability. The latter approach generalizes also to sums of $ d $-th powers of $k$-forms for $d \geq 3$ and $k \geq 2$., Comment: 14 pages. Code for the base case computations can be found on GitHub

We prove a conjecture due to Sturmfels and Uhler concerning the degree of the projective variety associated to the Gaussian graphical model of the cycle. We involve new methods based on the intersection theory in the space of complete quadrics.

Recently, we have witnessed tremendous applications of algebraic intersection theory to branches of mathematics, that previously seemed very distant. In this article we review some of them. Our aim is to provide a unified approach to the results e.g. in the theory of chromatic polynomials (work of Adiprasito, Huh, Katz), maximum likelihood degree in algebraic statistics (Drton, Manivel, Monin, Sturmfels, Uhler, Wi\'sniewski), Euler characteristics of determinental varieties (Dimca, Papadima), characteristic numbers (Aluffi, Schubert, Vakil) and the degree of semidefinite programming (Bothmer, Nie, Ranestad, Sturmfels). Our main tools come from intersection theory on special varieties called the varieties of complete forms (De Concini, Procesi, Thaddeus) and the study of Segre classes (Laksov, Lascoux, Pragacz, Thorup)., Comment: 37 pages; comments welcome!

We introduce the characteristic numbers and the chromatic polynomial of a tensor. Our approach generalizes and unifies the chromatic polynomial of a graph and of a matroid, characteristic numbers of quadrics in Schubert calculus, Betti numbers of complements of hyperplane arrangements and Euler characteristic of complements of determinantal hypersurfaces and the maximum likelihood degree for general linear concentration models in algebraic statistics.

We provide an algorithm that takes as an input a given parametric family of homogeneous polynomials, which is invariant under the action of the general linear group, and an integer $d$. It outputs the ideal of that family intersected with the space of homogeneous polynomials of degree $d$. Our motivation comes from open problems, which ask to find equations for varieties of cubic and quartic symmetroids. The algorithm relies on a database of specific Young tableaux and highest weight polynomials. We provide the database and the implementation of the database construction algorithm. Moreover, we provide a julia implementation to run the algorithm using the database, so that more varieties of homogeneous polynomials can easily be treated in the future.

We introduce the broad subclass of algebraic compressed sensing problems, where structured signals are modeled either explicitly or implicitly via polynomials. This includes, for instance, low-rank matrix and tensor recovery. We employ powerful techniques from algebraic geometry to study well-posedness of sufficiently general compressed sensing problems, including existence, local recoverability, global uniqueness, and local smoothness. Our main results are summarized in thirteen questions and answers in algebraic compressed sensing. Most of our answers concerning the minimum number of required measurements for existence, recoverability, and uniqueness of algebraic compressed sensing problems are optimal and depend only on the dimension of the model., Comment: 30 pages, 1 figure

The variety $ \mathrm{Sing}_{n, m} $ consists of all tuples $ X = (X_1,\ldots, X_m) $ of $ n\times n $ matrices such that every linear combination of $ X_1,\ldots, X_m $ is singular. Equivalently, $X\in\mathrm{Sing}_{n,m}$ if and only if $\det(\lambda_1 X_1 + \ldots + \lambda_m X_m) = 0$ for all $ \lambda_1,\ldots, \lambda_m\in \mathbb Q $. Makam and Wigderson asked whether the ideal generated by these equations is always radical, that is, if any polynomial identity that is valid on $ \mathrm{Sing}_{n, m} $ lies in the ideal generated by the polynomials $\det(\lambda_1 X_1 + \ldots + \lambda_m X_m)$. We answer this question in the negative by determining the vanishing ideal of $ \mathrm{Sing}_{2, m} $ for all $ m\in \mathbb N $. Our results exhibit that there are additional equations arising from the tensor structure of $ X $. More generally, for any $ n $ and $ m\ge n^2 - n + 1 $, we prove there are equations vanishing on $ \mathrm{Sing}_{n, m} $ that are not in the ideal generated by polynomials of type $\det(\lambda_1 X_1 + \ldots + \lambda_m X_m)$. Our methods are based on classical results about Fano schemes, representation theory and Gr\"obner bases., Comment: 11 pages. Sage Code for Gr\"obner bases included

We present three families of minimal border rank tensors: they come from highest weight vectors, smoothable algebras, or monomial algebras. We analyse them using Strassen's laser method and obtain an upper bound $2.431$ on $\omega$. We also explain how in certain monomial cases using the laser method directly is less profitable than first degenerating. Our results form possible paths in the search for valuable tensors for the laser method away from Coppersmith-Winograd tensors., Comment: v3: final v2: minor changes v1: 14 pages, comments welcome!

We prove that seminormality of cut polytopes is equivalent to normality. This settles two conjectures regarding seminormality of cut polytopes.

We establish connections between: the maximum likelihood degree (ML-degree) for linear concentration models, the algebraic degree of semidefinite programming (SDP), and Schubert calculus for complete quadrics. We prove a conjecture by Sturmfels and Uhler on the polynomiality of the ML-degree. We also prove a conjecture by Nie, Ranestad and Sturmfels providing an explicit formula for the degree of SDP. The interactions between the three fields shed new light on the asymptotic behaviour of enumerative invariants for the variety of complete quadrics. We also extend these results to spaces of general matrices and of skew-symmetric matrices., Comment: 35 pages, comments welcome

Whereas matrix rank is additive under direct sum, in 1981 Sch\"onhage showed that one of its generalizations to the tensor setting, tensor border rank, can be strictly subadditive for tensors of order three. Whether border rank is additive for higher order tensors has remained open. In this work, we settle this problem by providing analogs of Sch\"onhage's construction for tensors of order four and higher. Sch\"onhage's work was motivated by the study of the computational complexity of matrix multiplication; we discuss implications of our results for the asymptotic rank of higher order generalizations of the matrix multiplication tensor., Comment: 26 pages, 5 figures. Final version accepted in SIMAX

A toric variety is constructed from a lattice polytope. It is common in algebraic combinatorics to carry this way a notion of an algebraic property from the variety to the polytope. From the combinatorial point of view, one of the most interesting constructions of toric varieties comes from the base polytope of a matroid. Matroid base polytopes and independence polytopes are Cohen--Macaulay. We study two natural stronger algebraic properties -- Gorenstein and smooth. We provide a full classifications of matroids whose independence polytope or base polytope is smooth or Gorenstein. The latter answers to a question raised by Herzog and Hibi.

We study the maximum likelihood (ML) degree of linear concentration models in algebraic statistics. We relate it to an intersection problem on the variety of complete quadrics. This allows us to provide an explicit, basic, albeit of high computational complexity, formula for the ML-degree. The variety of complete quadrics is an exact analog for symmetric matrices of the permutohedron variety for the diagonal matrices., Comment: Minor changes

Wild forms are homogeneous polynomials whose smoothable rank is strictly larger than their border rank. The discrepancy between these two ranks is caused by the difference between the limit of spans of a family of zero-dimensional schemes and the span of their flat limit. For concise forms of minimal border rank, we show that the condition of vanishing Hessian is equivalent to being wild. This is proven by making a detour through structure tensors of smoothable and Gorenstein algebras. The equivalence fails in the non-minimal border rank regime. We exhibit an infinite series of minimal border rank wild forms of every degree $d\geq 3$ as well as an infinite series of wild cubics. Inspired by recent work on border apolarity of Buczy\'nska and Buczy\'nski, we study the border varieties of sums of powers $\underline{\mathrm{VSP}}$ of these forms in the corresponding multigraded Hilbert schemes., Comment: 25 pp., main result upgraded to an equivalence

We write down an explicit sequence of tensors in $C^m\otimes C^m\otimes C^m$, for all $m$ sufficiently large, having border rank at least $2.02m$, overcoming a longstanding barrier. We obtain our lower bounds via the border substitution method., Comment: 11 pages

Symmetric edge polytopes are a class of lattice polytopes constructed from finite simple graphs. In the present paper we highlight their connections to the Kuramoto synchronization model in physics -- where they are called adjacency polytopes -- and to Kantorovich--Rubinstein polytopes from finite metric space theory. Each of these connections motivates the study of symmetric edge polytopes of particular classes of graphs. We focus on such classes and apply algebraic-combinatorial methods to investigate invariants of the associated symmetric edge polytopes., Comment: 35 pages, 8 figures. Comments are very welcome!

Motivated by the study of the secant variety of the Segre-Veronese variety we propose a general framework to analyze properties of the secant varieties of toric embeddings of affine spaces defined by simplicial complexes. We prove that every such secant is toric, which gives a way to use combinatorial tools to study singularities. We focus on the Segre-Veronese variety for which we completely classify their secants that give Gorenstein or $\mathbb Q$-Gorenstein varieties. We conclude providing the explicit description of the singular locus.

Complete monotonicity is a strong positivity property for real-valued functions on convex cones. It is certified by the kernel of the inverse Laplace transform. We study this for negative powers of hyperbolic polynomials. Here the certificate is the Riesz kernel in Garding's integral representation. The Riesz kernel is a hypergeometric function in the coefficients of the given polynomial. For monomials in linear forms, it is a Gel'fand-Aomoto hypergeometric function, related to volumes of polytopes. We establish complete monotonicity for sufficiently negative powers of elementary symmetric functions. We also show that small negative powers of these polynomials are not completely monotone, proving one direction of a conjecture by Scott and Sokal., Comment: 22 pages

We show that the empirical risk minimization (ERM) problem for neural networks has no solution in general. Given a training set $s_1, \dots, s_n \in \mathbb{R}^p$ with corresponding responses $t_1,\dots,t_n \in \mathbb{R}^q$, fitting a $k$-layer neural network $\nu_\theta : \mathbb{R}^p \to \mathbb{R}^q$ involves estimation of the weights $\theta \in \mathbb{R}^m$ via an ERM: \[ \inf_{\theta \in \mathbb{R}^m} \; \sum_{i=1}^n \lVert t_i - \nu_\theta(s_i) \rVert_2^2. \] We show that even for $k = 2$, this infimum is not attainable in general for common activations like ReLU, hyperbolic tangent, and sigmoid functions. A high-level explanation is like that for the nonexistence of best rank-$r$ approximations of higher-order tensors --- the set of parameters is not a closed set --- but the geometry involved for best $k$-layer neural networks approximations is more subtle. In addition, we show that for smooth activations $\sigma(x)= 1/\bigl(1 + \exp(-x)\bigr)$ and $\sigma(x)=\tanh(x)$, such failure to attain an infimum can happen on a positive-measured subset of responses. For the ReLU activation $\sigma(x)=\max(0,x)$, we completely classifying cases where the ERM for a best two-layer neural network approximation attains its infimum. As an aside, we obtain a precise description of the geometry of the space of two-layer neural networks with $d$ neurons in the hidden layer: it is the join locus of a line and the $d$-secant locus of a cone., Comment: 19 pages

The toric variety of a matroid is projectively normal, and therefore it is Cohen-Macaulay. We provide a complete graph-theoretic classification when the toric variety of a graphic matroid is Gorenstein.

We apply methods from algebraic geometry to study uniform matrix product states. Our main results concern the topology of the locus of tensors expressed as uMPS, their defining equations and identifiability. By an interplay of theorems from algebra, geometry and quantum physics we answer several questions and conjectures posed by Critch, Morton and Hackbusch., Comment: 24 pages

In stochastic analysis, a standard method to study a path is to work with its signature. This is a sequence of tensors of different order that encode information of the path in a compact form. When the path varies, such signatures parametrize an algebraic variety in the tensor space. The study of these signature varieties builds a bridge between algebraic geometry and stochastics, and allows a fruitful exchange of techniques, ideas, conjectures and solutions. In this paper we study the signature varieties of two very different classes of paths. The class of rough paths is a natural extension of the class of piecewise smooth paths. It plays a central role in stochastics, and its signature variety is toric. The class of axis-parallel paths has a peculiar combinatoric flavour, and we prove that it is toric in many cases., Comment: Code for the computations is available at https://sites.google.com/view/l-colmenarejo/publications/code

We prove boundedness results for the injectivity regions for PEPS. Our result is a higher-dimensional generalization of the quantum Wielandt inequality., Comment: 6 pages

Matroids are ubiquitous in modern combinatorics. As discovered by Gelfand, Goresky, MacPherson and Serganova there is a beautiful connection between matroid theory and the geometry of Grassmannians: realizable matroids correspond to torus orbits in Grassmannians. Further, as observed by Fink and Speyer general matroids correspond to classes in the $K$-theory of Grassmannians. This yields in particular a geometric description of the Tutte polynomial. In this review we describe all these constructions in detail, and moreover we generalise some of them to polymatroids. More precisely, we study the class of flag matroids and their relations to flag varieties. In this way, we obtain an analogue of the Tutte polynomial for flag matroids.

Consider a linear space L of complex D-dimensional linear operators, and assume that some power L^k of L is the whole space of DxD matrices. Perez-Garcia, Verstraete, Wolf and Cirac conjectured that the sequence L^1,L^2,... stablilizes after O(D^2) terms; we prove that this happens after O(D^2 log(D)) terms, improving the previously known bound of O(D^4).

The goal of this article is to compare the coefficients in the expansion of the permanent with those in the expansion of the determinant of a three-lines circulant matrix. As an application we prove a conjecture concerning the minimality of Galois-Togliatti systems.

We investigate arithmetic, geometric and combinatorial properties of symmetric edge polytopes. We give a complete combinatorial description of their facets. By combining Gr\"obner basis techniques, half-open decompositions and methods for interlacing polynomials we provide an explicit formula for the $h^\ast$-polynomial in case of complete bipartite graphs. In particular, we show that the $h^\ast$-polynomial is $\gamma$-positive and real-rooted. This proves Gal's conjecture for arbitrary flag unimodular triangulations in this case, and, beyond that, we prove a strengthing due to Nevo and Petersen (2011)., Comment: 18 pages, 1 figure. Comments are welcome

In 1997 Oda conjectured that every smooth lattice polytope has the integer decomposition property. We prove Oda's conjecture for centrally symmetric $3$-dimensional polytopes, by showing they are covered by lattice parallelepipeds and unimodular simplices., Comment: 7 pages, 3 figures; revised version following the helpful comments of the referees

The image of a polynomial map is a constructible set. While computing its closure is standard in computer algebra systems, a procedure for computing the constructible set itself is not. We provide a new algorithm, based on algebro-geometric techniques, addressing this problem. We also apply these methods to answer a question of W. Hackbusch on the non-closedness of site-independent cyclic matrix product states for infinitely many parameters., Comment: 18 pages

We show that in finite-dimensional nonlinear approximations, the best $r$-term approximant of a function $f$ almost always exists over $\mathbb{C}$ but that the same is not true over $\mathbb{R}$, i.e., the infimum $\inf_{f_1,\dots,f_r \in Y} \lVert f - f_1 - \dots - f_r \rVert$ is almost always attainable by complex-valued functions $f_1,\dots, f_r$ in $Y$, a set of functions that have some desired structures. Our result extends to functions that possess special properties like symmetry or skew-symmetry under permutations of arguments. For the case where $Y$ is the set of separable functions, the problem becomes that of best rank-$r$ tensor approximations. We show that over $\mathbb{C}$, any tensor almost always has a unique best rank-$r$ approximation. This extends to other notions of tensor ranks such as symmetric rank and alternating rank, to best $r$-block-terms approximations, and to best approximations by tensor networks. When applied to sparse-plus-low-rank approximations, we obtain that for any given $r$ and $k$, a general tensor has a unique best approximation by a sum of a rank-$r$ tensor and a $k$-sparse tensor with a fixed sparsity pattern; this arises in, for example, estimation of covariance matrices of a Gaussian hidden variable model with $k$ observed variables conditionally independent given $r$ hidden variables. The existential (but not the uniqueness) part of our result also applies to best approximations by a sum of a rank-$r$ tensor and a $k$-sparse tensor with no fixed sparsity pattern, as well as to tensor completion problems., Comment: 25 pages, 4 figures

Can an ideal I in a polynomial ring k[x] over a field be moved by a change of coordinates into a position where it is generated by binomials $x^a - cx^b$ with c in k, or by unital binomials (i.e., with c = 0 or 1)? Can a variety be moved into a position where it is toric? By fibering the G-translates of I over an algebraic group G acting on affine space, these problems are special cases of questions about a family F of ideals over an arbitrary base B. The main results in this general setting are algorithms to find the locus of points in B over which the fiber of F - is contained in the fiber of a second family F' of ideals over B; - defines a variety of dimension at least d; - is generated by binomials; or - is generated by unital binomials. A faster containment algorithm is also presented when the fibers of F are prime. The big-fiber algorithm is probabilistic but likely faster than known deterministic ones. Applications include the setting where a second group T acts on affine space, in addition to G, in which case algorithms compute the set of G-translates of I - whose stabilizer subgroups in T have maximal dimension; or - that admit a faithful multigrading by $Z^r$ of maximal rank r. Even with no ambient group action given, the final application is an algorithm to - decide whether a normal projective variety is abstractly toric. All of these loci in B and subsets of G are constructible; in some cases they are closed., Comment: 22 pages

The technical details of a balloon stratospheric mission that is aimed at measuring the Schumann resonances are described. The gondola is designed specifically for the measuring of faint effects of ELF (Extremely Low Frequency electromagnetic waves) phenomena. The prototype met the design requirements. The ELF measuring system worked properly for entire mission; however, the level of signal amplification that was chosen taking into account ground-level measurements was too high. Movement of the gondola in the Earth magnetic field induced the signal in the antenna that saturated the measuring system. This effect will be taken into account in the planning of future missions. A large telemetry dataset was gathered during the experiment and is currently under processing. The payload consists also of biological material as well as electronic equipment that was tested under extreme conditions., Comment: 13 pages, 9 figures