Identical Vanishing Research Articles

Exploring the dynamics of collisionless spherical stellar systems using the matrix method: Insights from the dilation mode

Context. Analytical solutions to the perturbed equations that govern self-gravitating collisionless stellar systems are crucial for both code testing and theoretical insights. For spheres, a solution has been known for years that corresponds to the entire object’s shift from the origin. We recently introduced a new exact stationary solution, relevant for models with a single length parameter. This solution, referred to as the scale-invariant or dilation mode, has led to insights regarding the concept of perturbation energy within the linear theory framework. Aims. Our aim is to use Hénon’s isochrone model as an example to verify the ability of the standard matrix method to successfully predict the existence of a dilation mode, and to explore its potential application as a test disturbance. Methods. We used the standard matrix method for radial perturbations and applied Clutton-Brock potential-density pairs to determine the properties of the perturbations. Results. In this particular case of stationary radial perturbations, the typical relationship between the perturbations of the distribution function and the potential fails. This discrepancy poses a challenge when attempting to use the dilation mode as a test. When using Clutton-Brock pairs with the matrix method, a mass conservation equation as an additional equation to the ordinary set of linear equations is required. With this added equation, it’s possible to obtain the needed test: identical vanishing of the determinant of this modified set of equations with an increasing number of included basis functions.

Rigid Biholomorphic Equivalences of Rigid C2,1 Hypersurfaces M5⊂C3

We study the local equivalence problem for real-analytic (Cω) hypersurfaces M5⊂C3 that, in some holomorphic coordinates (z1,z2,w)∈C3 with w=u+ −1v, are rigid in the sense that their graphing functions u=F(z1,z2,z‾1,z‾2) are independent of v. Specifically, we study the group Holrigid(M) of rigid local biholomorphic transformations of the form (z1,z2,w)⟼(f1(z1,z2),f2(z1,z2),aw+g(z1,z2)), where a∈R∖{0} and D(f1,f2)/D(z1,z2)≠0, which preserve the rigidity of hypersurfaces. After performing a Cartan-type reduction to an appropriate {e}-structure, we find exactly two primary invariants I0 and V0, which we express explicitly in terms of the 5-jet of the graphing function F of M. The identical vanishing 0≡I0(J5F)≡V0(J5F) then provides a necessary and sufficient condition for M to be locally rigidly biholomorphic to the known model hypersurface MLC:u=z1z‾1+1/2z12z‾2+1/2z‾12z2 1−z2z‾2. We establish that always dimHolrigid(M)≤7=dimHolrigid(MLC). If one of these two primary invariants I0≢0 or V0≢0 does not vanish identically, then on either of the two Zariski-open sets {p∈M:I0(p)≠0} or {p∈M:V0(p)≠0}, we show that this rigid equivalence problem between rigid hypersurfaces reduces to an equivalence problem for a certain five-dimensional {e}-structure on M, that is, we get an invariant absolute parallelism on M5. Hence dimHolrigid(M) drops from 7 to 5, illustrating the gap phenomenon.

Symmetrically Colored Gaussian Graphical Models with Toric Vanishing Ideals

A colored Gaussian graphical model is a linear concentration model in which equalities among the concentrations are specified by a coloring of an underlying graph. The model is called RCOP if this coloring is given by the edge and vertex orbits of a subgroup of the automorphism group of the graph. We show that RCOP Gaussian graphical models on block graphs are toric in the space of covariance matrices and we describe Markov bases for them. To this end, we learn more about the combinatorial structure of these models and their connection to Jordan algebras.

Ideal Membership Problem over 3-Element CSPs with Dual Discriminator Polymorphism

In this paper we examine polynomial ideals that are the vanishing ideals of solution sets of combinatorial problems encoded by constraint satisfaction problems over a finite language. We consider a 3-element domain and the dual discriminator polymorphism (constraints under this polymorphism are a generalization of the 2-satisfiability problem). Assuming the graded lexicographic ordering of monomials, we show that the reduced Gröbner basis of ideals whose varieties are closed under this polymorphism can be computed in polynomial time. This proves polynomial time solvability of the ideal membership problem (IMP) with restrictions on degree $d=O(1)$, which we call IMP$_d$, for these constrained problems. It is a first step toward the challenging long-term goal of identifying when IMP$_d$ is polynomial time solvable for a finite domain.

Geometric Generalized Wronskians: Applications in Intermediate Hyperbolicity and Foliation Theory

Abstract In this paper, we introduce a sub-family of the usual generalized Wronskians, which we call geometric generalized Wronskians. It is well known that one can test linear dependance of holomorphic functions (of several variables) via the identical vanishing of generalized Wronskians. We show that such a statement remains valid if one tests the identical vanishing only on geometric generalized Wronskians. It turns out that geometric generalized Wronskians allow to define intrinsic objects on projective varieties polarized with an ample line bundle: in this setting, the lack of existence of global functions is compensated by global sections of powers of the fixed ample line bundle. Geometric generalized Wronskians are precisely defined so that their local evaluations on such global sections globalize up to a positive twist by the ample line bundle. We then give three applications of the construction of geometric generalized Wronskians: one in intermediate hyperbolicity and two in foliation theory. In intermediate hyperbolicity, we show the algebraic degeneracy of holomorphic maps from ${\mathbb {C}}^{p}$ to a Fermat hypersurface in $\textbf {P}^{N}$ of degree $\delta> (N+1)(N-p)$: this interpolates between two well-known results, namely for $p=1$ (first proved via Nevanlinna theory) and $p=N-1$ (in which case the Fermat hypersurface is of general type). The first application in foliation theory provides a criterion for algebraic integrability of leaves of foliations: our criterion is not optimal in view of current knowledge, but has the advantage of having an elementary proof. Our second application deals with positivity properties of adjoint line bundles of the form $K_{\mathcal {F}} + L $, where $K_{\mathcal {F}}$ is the canonical bundle of a regular foliation $\mathcal {F}$ on a smooth projective variety $X$, and where $L$ is an ample line bundle on $X$.

Directed Gaussian graphical models with toric vanishing ideals

Directed Gaussian graphical models are statistical models that use a directed acyclic graph (DAG) to represent the conditional independence structures between a set of jointly normal random variables. The DAG specifies the model through recursive factorization of the parametrization, via restricted conditional distributions. In this paper, we make an attempt to characterize the DAGs whose vanishing ideals are toric ideals. In particular, we give some combinatorial criteria to construct such DAGs from smaller DAGs which have toric vanishing ideals. An associated monomial map called the shortest trek map plays an important role in our description of toric Gaussian DAG models. For DAGs whose vanishing ideal is toric, we prove results about the generating sets of those toric ideals.

Gaussian graphical models with toric vanishing ideals

Gaussian graphical models are semi-algebraic subsets of the cone of positive definite covariance matrices. They are widely used throughout natural sciences, computational biology and many other fields. Computing the vanishing ideal of the model gives us an implicit description of the model. In this paper, we resolve two conjectures given by Sturmfels and Uhler. In particular, we characterize those graphs for which the vanishing ideal of the Gaussian graphical model is generated in degree 1 and 2. These turn out to be the Gaussian graphical models whose ideals are toric ideals, and the resulting graphs are the 1-clique sums of complete graphs.

Spurious Vanishing Problem in Approximate Vanishing Ideal

Approximate vanishing ideal is a concept from computer algebra that studies the algebraic varieties behind perturbed data points. To capture the nonlinear structure of perturbed points, the introduction of approximation to exact vanishing ideals plays a critical role. However, such an approximation also gives rise to a theoretical problem-the spurious vanishing problem-in the basis construction of approximate vanishing ideals; namely, obtained basis polynomials can be approximately vanishing simply because of the small coefficients. In this paper, we propose a first general method that enables various basis construction algorithms to overcome the spurious vanishing problem. In particular, we integrate coefficient normalization with polynomial-based basis constructions, which do not need the proper ordering of monomials to process for basis constructions. We further propose a method that takes advantage of the iterative nature of basis construction so that computationally costly operations for coefficient normalization can be circumvented. Moreover, a coefficient truncation method is proposed for further accelerations. From the experiments, it can be shown that the proposed method overcomes the spurious vanishing problem, resulting in shorter feature vectors while sustaining comparable or even lower classification error.

Matrix Schubert varieties and Gaussian conditional independence models

Matrix Schubert varieties are certain varieties in the affine space of square matrices which are determined by specifying rank conditions on submatrices. We study these varieties for generic matrices, symmetric matrices, and upper triangular matrices in view of two applications to algebraic statistics: We observe that special conditional independence models for Gaussian random variables are intersections of matrix Schubert varieties in the symmetric case. Consequently, we obtain a combinatorial primary decomposition algorithm for some conditional independence ideals. We also characterize the vanishing ideals of Gaussian graphical models for generalized Markov chains. In the course of this investigation, we are led to consider three related stratifications, which come from the Schubert stratification of a flag variety. We provide some combinatorial results, including describing the stratifications using the language of rank arrays and enumerating the strata in each case.

Generalized non-commutative degeneration conjecture

We propose a generalization of the Kontsevich–Soibelman conjecture on the degeneration of the Hochschild-to-cyclic spectral sequence for a smooth compact differential graded category. Our conjecture states identical vanishing of a certain map between bi-additive invariants of arbitrary small differential graded categories over a field of characteristic zero. We show that this generalized conjecture follows from the Kontsevich–Soibelman conjecture and the so-called conjecture on smooth categorical compactification.

Binomial vanishing ideals

In this note we characterize, in algebraic and geometric terms, when a graded vanishing ideal is generated by binomials over any field K.

Parameterized affine codes

Let K be a finite field and let X* be an affine algebraic toric set parameterized by monomials. We give an algebraic method, using Gröbner bases, to compute the length and the dimension of CX* (d), the parameterized affine code of degree d on the set X*. If Y is the projective closure of X*, it is shown that CX* (d) has the same basic parameters that CY (d), the parameterized projective code on the set Y. If X* is an affine torus, we compute the basic parameters of CX* (d). We show how to compute the vanishing ideals of X* and Y.

Generating Invariants for Hybrid Systems by Computing Vanishing Ideals of Sample Points

In this paper, we investigate the problem of generating invariants of hybrid systems. We present a new approach, for generating polynomial equation invariants of hybrid systems through computing vanishing ideal of sample points. The challenge is to deal with the continuous consecution of hybrid systems because of the existence of differential equations. Our approach avoids first-order quantifier elimination and cylindrical algebraic decomposition (CAD). From the preliminary experiment results, we demonstrate the feasibility of our approach.

Linear dependence of quotients of analytic functions of several variables with the least subcollection of generalized Wronskians

We study linear dependence in the case of quotients of analytic functions in several variables (real or complex). We identify the least subcollection of generalized Wronskians whose identical vanishing is sufficient for linear dependence. Our proof admits a straight-forward algebraic generalization and also constitutes an alternative proof of the previously known result that the identical vanishing of the whole collection of generalized Wronskians implies linear dependence. Motivated by the structure of this proof, we introduce a method for calculating the space of linear relations. We conclude with some reflections about this method that may be promising from a computational point of view.

An Alternative Form of the Helmholtz Criterion in the Inverse Problem of Calculus of Variations

We give a necessary and sufficient condition for the existence of a local solution of the inverse problem of calculus of variations in terms of the identical vanishing of the variation of a functional on an extended space (with the number of independent variables increased by one), and explain its relation with the classical Helmholtz criterion using the de Rham complex on an infinite-dimensional space of fields.

High-order vanishing ideals at n+3 points of Pn

The authors conjecture that the ideal IZ[t−1] of functions vanishing to order t−1 at the subscheme Z of Pn, n=2t−1 comprised of 2t+2 generic smooth points, satisfies dimk(IZ[t−1])t=2t, in its initial degree, t. They show that this dimension is at least t+1, by a direct construction of suitable vanishing forms. This result is complementary to those of M.V. Catalisano, P. Ellia, and A. Gimigliano in [5]. The authors also consider related problems, including the Macaulay dual problem, of determining the Hilbert function H(A), A=R/(x12,…,xr2,(x1+⋯+xr)2,L2) — where R=k[x1,…,xr], r=2t and L is a generic linear form — in the socle degree t of A.

Generic Transmission Zeros and In-Phase Resonances in Time-Reversal Symmetric Single Channel Transport

We study phase coherent transport in a single channel system using the scattering matrix approach. It is shown that identical vanishing of the transmission amplitude occurs generically in quasi-1D systems if the time-reversal is a good symmetry. The transmission zeros naturally lead to abrupt phase changes (without any intrinsic energy scale) and in-phase resonances, providing insights to recent experiments on phase coherent transport through a quantum dot.

On ball's mapping of the two-system of screws

Ball's circular representation of the two-system of screws is generalized using a new formulation based on the eliminant of two quadratic equations. Special two-systems, which correspond to degenerate forms of the cylindroid, are systematically determined by the identical vanishing of the eliminant. Further, an original definition by Clifford is extended to Euclidean space to elegantly characterize the pitch and axis of a screw via projective metrics.

Problem of the identical vanishing of Euler-Lagrange derivatives in field theory

I prove that the necessary and sufficient condition for two Lagrangian densities ${\mathcal{L}}_{1}({\ensuremath{\psi}}^{A};{{\ensuremath{\psi}}^{A}}_{,\ensuremath{\alpha}})$ and ${\mathcal{L}}_{2}({\ensuremath{\psi}}^{A};{{\ensuremath{\psi}}^{A}}_{,\ensuremath{\alpha}})$ to have exactly the same Euler-Lagrange derivatives is that their difference $\ensuremath{\Delta}({\ensuremath{\psi}}^{A};{{\ensuremath{\psi}}^{A}}_{,\ensuremath{\alpha}})$ be the divergence of ${\ensuremath{\omega}}^{\ensuremath{\mu}}({\ensuremath{\psi}}^{A};{{\ensuremath{\psi}}^{A}}_{,\ensuremath{\alpha}};{x}^{\ensuremath{\mu}})$ with a given dependence on ${{\ensuremath{\psi}}^{A}}_{,\ensuremath{\alpha}}$. The main point is that ${\ensuremath{\omega}}^{\ensuremath{\mu}}$ depends on ${{\ensuremath{\psi}}^{A}}_{,\ensuremath{\alpha}}$ but $\ensuremath{\Delta}$ does not depend on second derivatives of the field ${\ensuremath{\psi}}^{A}$. Therefore, the function $\ensuremath{\Delta}$ need not be linear in ${{\ensuremath{\psi}}^{A}}_{,\ensuremath{\alpha}}$.

Covariants of pencils of binary cubics

SynopsisThe invariants of pencils of binary cubics were described in an earlier paper by the author. In this note, we give a complete set of covariants for such pencils, and show how each type of pencil may be denned by the identical vanishing of covariants.