Abstract. Continuous spline functions are defined as piecewise polynomials on the faces of a polyhedral complex that agree on the intersections of two faces. Splines are used in approximation theory and numerical analysis, with applications in data interpolation, to create smooth curves in computer graphics and to find numerical solutions to partial differential equations. Gilbert, Tymoczko, and Viel [ Pacific J. Math., 281 (2016), pp. 333–364] generalized the classical splines combinatorially and algebraically: a generalized spline is a vertex labeling of a graph [Formula: see text] by elements of the ring so that the difference between the labels of any two adjacent vertices lies in the ideal generated by the corresponding edge label. We study the generalized splines on the planar graphs whose edges are labeled by two-variable polynomials of the form [Formula: see text] and whose vertices are labeled by polynomials of degree at most two. In this paper we, address the upper-bound conjecture for the dimension of degree-2 splines of smoothness 1. The dimension is expressed in terms of the rank of the extended cycle basis matrix. We also provide a combinatorial algorithm on graphs to compute the rank by contracting certain subgraphs.
Abstract. We consider the representations of the group [Formula: see text] that contain the standard representation [Formula: see text] as an irreducible component. This is a large class of representations containing the vector spaces of ternary forms of any odd degree, the space of piezoelectricity tensors, or the classical product space [Formula: see text] which appears in multiple physical and engineering situations. For any such representation, we construct a set of algebraically independent rational invariants that generates the invariant field. Our key ingredient is a constructive use of Seshadri slice lemma. This lemma establishes an isomorphism between the [Formula: see text]-invariant field on the full representation [Formula: see text] with the invariant field of a representation of [Formula: see text] on a subspace [Formula: see text]. We exhibit a set of [Formula: see text]-invariant polynomials generating this latter invariant field. The explicit expressions for the generating rational [Formula: see text]-invariants can then be obtained with symbolic manipulations. The results are then extended to actions of [Formula: see text] and several cases of interest to applications are detailed as examples.
Abstract. We consider the problem of computing matrix polynomials [Formula: see text], where [Formula: see text] is a large dense matrix, with as few matrix-matrix multiplications as possible. More precisely, let [Formula: see text] represent the set of polynomials computable with [Formula: see text] matrix-matrix multiplications, but with an arbitrary number of matrix additions and scaling operations. We characterize this set through a tabular parameterization. By deriving equivalence transformations of the tabular representation, we establish new methods that can be used to construct elements of [Formula: see text] and determine general properties of the set. The transformations allow us to eliminate variables and prove that the dimension is bounded by [Formula: see text], which is subsequently proven to be sharp, i.e., [Formula: see text]. Consequently, we have identified a parameterization that, to the best of our knowledge, is the first minimal parameterization. We also conduct a study using computational tools from algebraic geometry to determine the largest degree [Formula: see text] such that all polynomials of that degree belong to [Formula: see text] or its closure. In many cases, the computational setup is constructive in the sense that it can also be used to determine a specific evaluation scheme for a given polynomial.
We consider the representations of the group SO3(R) that contain the standard representation R3 as an irreducible component. This is a large class of representations containing the vector spaces of ternary forms of any odd degree, the space of piezoelectricity tensors, or the classical product space R3 imes \cdot \cdot \cdot imes R3 which appears in multiple physical and engineering situations. For any such representation, we construct a set of algebraically independent rational invariants that generates the invariant field. Our key ingredient is a constructive use of Seshadri slice lemma. This lemma establishes an isomorphism between the SO3(R)-invariant field on the full representation R3\oplus\scrH with the invariant field of a representation of O2(R) on a subspace R \oplus \scrH. We exhibit a set of O2(R)-invariant polynomials generating this latter invariant field. The explicit expressions for the generating rational SO3(R)-invariants can then be obtained with symbolic manipulations. The results are then extended to actions of O3(R) and several cases of interest to applications are detailed as examples.
We develop a discrete differential geometry for surfaces of non-constant negative curvature, which can be used to model various phenomena from the growth of flower petals to marine invertebrate swimming. Specifically, we derive and numerically integrate a version of the classical Lelieuvre formulas that apply to immersions of C1,1 hyperbolic surfaces of non-constant curvature. In contrast to the constant curvature case, these formulas do not provide an explicit method for constructing an immersion but rather describe an immersion via an implicit set of equations. We propose an iterative method for resolving these equations. Because we are interested in scenarios where the curvature is a function of the intrinsic material coordinates, in particular, on the geodesic distance from an origin or from an edge, we suggest a fast marching method for computing geodesic distance on manifolds. We apply our methods to generate surfaces of non-constant curvature and demonstrate how one can introduce branch points to account for the multi-generational buckling and subwrinkling observed in many applications.
We study statistical models that are parametrized by squares of linear forms. All critical points of the likelihood function are real and positive. There is one critical point in each region of the projective hyperplane arrangement defined by the linear forms. We examine the ideal and singular locus of the model, and we give a determinantal presentation for its likelihood correspondence. We characterize tropical degenerations of the maximum likelihood estimation (MLE), we describe the log-normal polytopes, and we explore connections to determinantal point processes.
We develop algebraic geometry for coupled cluster theory in second quantization. In quantum chemistry, electronic systems are represented by elements in the exterior algebra. The creation and annihilation operators of particles generate a Clifford algebra known as the Fermi-Dirac algebra. We present a non-commutative Gröbner basis giving an alternative proof of Wick's theorem, a foundational result in quantum chemistry. In coupled cluster theory, the Schrödinger equation is approximated through a hierarchy of polynomial equations at various levels of truncation. The exponential parameterization gives rise to the Fock space truncation varieties. This reveals well-known varieties, such as the Grassmannian, flag varieties and spinor varieties. We offer a detailed study of the truncation varieties and their CC degrees. We classify all cases for when the CC degree is equal to the degree of a graph of the exponential parametrization.
Minimal surfaces play a fundamental role in differential geometry, with applications spanning physics, material science, and geometric design. In this paper, we explore a novel quaternionic representation of minimal surfaces, drawing an analogy with the well-established theory of Pythagorean Hodograph (PH) curves. By exploiting the algebraic structure of complex quaternions, we introduce a new approach to generating minimal surfaces via quaternionic transformations. This method extends classical Weierstraß-Enneper representations and provides insights into the interplay between quaternionic analysis, PH curves, and minimal surface geometry. Additionally, we discuss the role of the Sylvester equation in this framework and demonstrate practical examples, including the construction of Enneper surface patches. The findings open new avenues in computational geometry and geometric modeling, bridging abstract algebraic structures with practical applications in CAD and computer graphics.
While the notion of isometric deformations of surfaces is straightforward for surfaces with Euclidean metric, a corresponding notion in isotropic space has been missing. By making Gauss' Theorema Egregium a necessary condition we develop a sensible notion of isometric surfaces in isotropic space. The well-known examples in Euclidean space, like isometries within the associated family of minimal surfaces, Bour's theorem, and Minding isometries, find their natural analogues in isotropic space. We also include an extensive treatment of infinitesimal flexibility, or infinitesimal deformation, of surfaces. We prove results for the isotropic displacement diagrams in analogy to its well-known counterparts in Euclidean space culminating in the existence of an isotropic Darboux wreath consisting of six surfaces. We show several interesting relations for special parametrizations involving Koenigs and Voss nets of smooth and discrete surfaces within the Darboux wreath and we encounter surfaces of constant Gaussian and mean curvature. At several occasions, we point to connections to statics as the isotropic space is a natural language to describe the Airy stress function.
We study shallow neural networks with monomial activations and output dimension one. The function space for these models can be identified with a set of symmetric tensors with bounded rank. We describe general features of these networks, focusing on the relationship between width and optimization. We then consider teacher-student problems, which can be viewed as problems of low-rank tensor approximation with respect to non-standard inner products that are induced by the data distribution. In this setting, we introduce a teacher-metric data discriminant which encodes the qualitative behavior of the optimization as a function of the training data distribution. Finally, we focus on networks with quadratic activations, presenting an in-depth analysis of the optimization landscape. In particular, we present a variation of the Eckart-Young Theorem characterizing all critical points and their Hessian signatures for teacher-student problems with quadratic networks and Gaussian training data.
When is it possible to project two sets of labeled points of equal cardinality lying in a pair of projective planes to the same image on a projective line? We give a complete answer to this question, obtaining the following results. We first show that such a pair of projections exist if and only if the two point sets are themselves images of a common point set in projective space. Moreover, we find that for generic pairs of point sets, a common projection exists if and only if their cardinality is at most seven. In these cases, we give an explicit description of the loci of projection centers that enable a common image.
The log canonical threshold (LCT) is a fundamental invariant in birational geometry, essential for understanding the complexity of singularities in algebraic varieties. Its real counterpart, the real log canonical threshold (RLCT), also known as the learning coefficient, has become increasingly relevant in statistics and machine learning, where it plays a critical role in model selection and error estimation for singular statistical models. In this paper, we investigate the RLCT and its multiplicity for real (not necessarily reduced) hyperplane arrangements. We derive explicit combinatorial formulas for these invariants, generalizing earlier results that were limited to specific examples. Moreover, we provide a general algebraic theory for RLCTs and present a SageMath implementation for efficiently computing the RLCT and its multiplicity in the case of real hyperplane arrangements. Applications to examples are given, illustrating how the formulas can also be used to analyze the asymptotic behavior of high-dimensional volume integrals.
We introduce a new technique to construct rank-metric codes using the arithmetic theory of Drinfeld modules over global fields, and Dirichlet Theorem on polynomial arithmetic progressions. Using our methods, we obtain a new infinite family of optimal rank-metric codes with rank-locality, i.e. every code in our family achieves the information theoretical bound for rank-metric codes with rank-locality.
We consider a new multivariate generalization of the classical monic (univariate) Chebyshev polynomial that minimizes the uniform norm on the interval [-1,1]. Let Π^*_n be the subset of polynomials of degree at most n in d variables, whose homogeneous part of degree n has coefficients summing up to 1. The problem is determining a polynomial in Π^*_n with the smallest uniform norm on a domain Ω, which we call a least Chebyshev polynomial (associated with Ω). Our main result solves the problem for Ω belonging to a non-trivial class of sets that we call diagonally-determined, and establishes the remarkable result that a least Chebyshev polynomial can be given via the classical, univariate, Chebyshev polynomial. In particular, the solution can be independent of the dimension. Diagonally-determined domains include centered balls in ℝ^d in any norm, but can be non-convex and even non-simply connected. We also introduce a computational procedure, based on semidefinite programming hierarchies, to detect if a given semi-algebraic set is diagonally-determined.
The field of numerical algebraic geometry consists of algorithms for numerically solving systems of polynomial equations. When the system is exact, such as having rational coefficients, the solution set is well-defined. However, for a member of a parameterized family of polynomial systems where the parameter values may be measured with imprecision or arise from prior numerical computations, uncertainty may arise in the structure of the solution set, including the number of isolated solutions, the existence of higher dimensional solution components, and the number of irreducible components along with their multiplicities. The loci where these structures change form a stratification of exceptional algebraic sets in the space of parameters. We describe methodologies for making the interpretation of numerical results more robust by searching for nearby parameter values on an exceptional set. We demonstrate these techniques on several illustrative examples and then treat several more substantial problems arising from the kinematics of mechanisms and robots.
On Hadamard manifolds, the radial fields, which are the negative gradients of the Busemann functions, can be used to designate a canonical sense of direction. This could have many potential applications to Hadamard manifold-valued data, for example in defining notions of quantiles or treatment effects. Some of the most commonly encountered Hadamard manifolds in statistics are the spaces of symmetric positive definite matrices, which are used in, for example, covariance matrix analysis and diffusion tensor imaging. Surprisingly, an expression for the radial fields on these manifolds is unavailable in the literature even though the issue arises quite naturally when studying the geometry of these spaces. This paper aims to fill this gap by deriving such an expression, and also demonstrates their smoothness.
We study the last fall degrees of {\em semi-local} polynomial systems, and the computational complexity of solving such systems for closed-point and rational-point solutions, where the systems are defined over a finite field. A semi-local polynomial system specifies an algebraic set which is the image of a global linear transformation of a direct product of local affine algebraic sets. As a special but interesting case, polynomial systems that arise from Weil restriction of algebraic sets in an affine space of low dimension are semi-local. Such systems have received considerable attention due to their application in cryptography. Our main results bound the last fall degree of a semi-local polynomial system in terms of the number of closed point solutions, and yield an efficient algorithm for finding all rational-point solutions when the prime characteristic of the finite field and the number of rational solutions are small. Our results on solving semi-local systems imply an improvement on a previously known polynomial-time attack on the HFE (Hidden Field Equations) cryptosystems. The attacks implied in our results extend to public key encryption functions which are based on semi-local systems where either the number of closed point solutions is small, or the characteristic of the field is small. It remains plausible to construct public key cryptosystems based on semi-local systems over a finite field of large prime characteristic with exponential number of closed point solutions. Such a method is presented in the paper, followed by further cryptanalysis involving the isomorphism of polynomials (IP) problem, as well as a concrete public key encryption scheme which is secure against all the attacks discussed in this paper.
We import the algebro-geometric notion of a complete collineation into the study of maximum likelihood estimation in directed Gaussian graphical models. A complete collineation produces a perturbation of sample data, which we call a stabilisation of the sample. While a maximum likelihood estimate (MLE) may not exist or be unique given sample data, it is always unique given a stabilisation. We relate the MLE given a stabilisation to the MLE given original sample data, when one exists, providing necessary and sufficient conditions for the MLE given a stabilisation to be one given the original sample. For linear regression models, we show that the MLE given any stabilisation is the minimal norm choice among the MLEs given an original sample. We show that the MLE has a well-defined limit as the stabilisation of a sample tends to the original sample, and that the limit is an MLE given the original sample, when one exists. Finally, we study which MLEs given a sample can arise as such limits. We reduce this to a question regarding the non-emptiness of certain algebraic varieties.
The motivation for this paper is to detect when an irreducible projective variety V is not toric. We do this by analyzing a Lie group and a Lie algebra associated to V. If the dimension of V is strictly less than the dimension of the above mentioned objects, then V is not a toric variety. We provide an algorithm to compute the Lie algebra of an irreducible variety and use it to provide examples of non-toric statistical models in algebraic statistics.
We develop a discrete differential geometry for surfaces of non-constant negative curvature, which can be used to model various phenomena from the growth of flower petals to marine invertebrate swimming. Specifically, we derive and numerically integrate a version of the classical Lelieuvre formulas that apply to immersions of $C^{1,1}$ hyperbolic surfaces of non-constant curvature. In contrast to the constant curvature case, these formulas do not provide an explicit method for constructing an immersion but rather describe an immersion via an implicit set of equations. We propose an iterative method for resolving these equations. Because we are interested in scenarios where the curvature is a function of the intrinsic material coordinates, in particular, on the geodesic distance from an origin, we suggest a fast marching method for computing geodesic distance on manifolds. We apply our methods to generate surfaces of non-constant curvature and demonstrate how one can introduce branch points to account for the sort of multi-generational buckling and subwrinkling observed in many applications.
