arXiv subjects
Publications and source records attributed to Paul Breiding.
This article introduces a new implementation for computing a numerical irreducible decomposition for a system of polynomial equations. The implementation is part of the software package HomotopyContinuation.jl written in the programming language Julia.
We introduce a low-memory framework for certifying numerical solutions to polynomial systems which uses solution iterators and spatial partitioning trees to reduce memory requirements. We provide a prototypical algorithm, analyze its complexity, and demonstrate the memory reduction on a large example.
Many hypersurfaces in algebraic geometry, such as discriminants, arise as the projection of another variety. The real complement of such a hypersurface decomposes into connected components. In this paper, we propose a new method for computing these components. Existing methods require the explicit equation of the hypersurface as input. However, computing this equation by elimination can be computationally demanding or even infeasible. Our approach instead derives from univariate interpolation by computing the intersection of the hypersurface with a line. Such an intersection may be computed using so-called pseudo-witness sets without computing a defining equation for the hypersurface. We implement our approach in a forthcoming Julia package and demonstrate, on several examples, that the resulting algorithm accurately recovers all components of the real complement of the hypersurface.
We study the critical points over an algebraic variety of an optimization problem defined by a quadratic objective that is degenerate. This scenario arises in machine learning when the dataset size is small with respect to the model, and is typically referred to as overparametrization. Our main result relates the degenerate optimization problem to a nondegenerate one via a projection. In the highly-degenerate regime, we find that a central role is played by the ramification locus of the projection. Additionally, we provide tools for counting the number of critical points over projective varieties, and discuss specific cases arising from deep learning. Our work bridges tools from algebraic geometry with ideas from machine learning, and it extends the line of literature around the Euclidean distance degree to the degenerate setting.
We introduce the concept of homotopy iterators for performing polynomial homotopy continuation tasks in a memory efficient manner. The main idea is to push forward an iterator for the start solutions of a homotopy via the function which tracks them along the homotopy. Doing so produces a representation of the target solutions, bypassing the need to hold all solutions in memory. We discuss several applications of this datatype ranging from solution counting to data compression.
Let $M=G/H$ be a Riemannian homogeneous space, where $G$ is a compact Lie group with closed subgroup $H$. Classical intersection theory states that the de Rham cohomology ring of $M$ describes the signed count of intersection points of submanifolds $Y_1, \ldots, Y_s$ of $M$ in general position, when the codimensions add up to $\dim M$. We introduce the probabilistic intersection ring $\mathrm{H}_{\mathbb E}(M)$, whose multiplication describes the unsigned count of intersection points, when the $Y_i$ are randomly moved by independent uniformly random elements of $G$. The probabilistic intersection ring $\mathrm{H}_{\mathbb E}(M)$ has the structure of a graded commutative and associative real Banach algebra. It is defined as a quotient of the ring of Grassmann zonoids of a fixed cotangent space $V$ of $M$. The latter was introduced by the authors in [Adv. Math. 402, 2022]. There is a close connection to valuations of convex bodies: $\mathrm{H}_{\mathbb E}(M)$ can be interpreted as a subspace of the space of translation invariant, even, continuous valuations on $V$, whose multiplication coincides with Alesker's multiplication for smooth valuations. We describe the ring structure of the probabilistic intersection ring for spheres, real projective space and complex projective space, relying on Fu [J. Diff. Geo. 72(3), 2006] for the latter case. From this, we derive an interesting probabilistic intersection formula in complex projective space. Finally, we initiate the investigation of the probabilistic intersection ring for real Grassmannians, outlining the construction of a probabilistic version of Schubert Calculus.
We present a Julia package HypersurfaceRegions.jl for computing all connected components in the complement of an arrangement of real algebraic hypersurfaces in $\mathbb{R}^n$.
In this paper we study typical ranks of real $m\times n \times \ell$ tensors. In the case $ (m-1)(n-1)+1 \leq \ell \leq mn$ the typical ranks are contained in $\{\ell, \ell +1\}$, and $\ell$ is always a typical rank. We provide a geometric proof of this fact. We express the probabilities of these ranks in terms of the probabilities of the numbers of intersection points of a random linear space with the Segre variety. In addition, we give some heuristics to understand how the probabilities of these ranks behave, based on asymptotic results on the average number of real points in a random linear slice of a Segre variety with a subspace of complementary dimension. The typical ranks of real $3\times 3\times 5$ tensors are $5$ and $6$. We link the rank probabilities of a $3\times 3 \times 5$ tensor with i.i.d.\ Gaussian entries to the probability of a random cubic surface in $\P^3$ having real lines. As a consequence, we get a bound on the expected number of real lines on such a surface.
Subspace varieties are algebraic varieties whose elements are tensors with bounded multilinear rank. In this paper, we compute their degrees by computing their volumes.
We compute the reach, extremal curvature and volume of a tubular neighborhood for the Segre-Veronese variety intersected with the unit sphere.
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 study the following problem in computer vision from the perspective of algebraic geometry: Using $m$ pinhole cameras we take $m$ pictures of a line in $\mathbb P^3$. This produces $m$ lines in $\mathbb P^2$ and the question is which $m$-tuples of lines can arise that way. We are interested in polynomial equations and therefore study the complex Zariski closure of all such tuples of lines. The resulting algebraic variety is a subvariety of $(\mathbb P^2)^m$ and is called line multiview variety. In this article, we study its ideal. We show that for generic cameras the ideal is generated by $3\times 3$-minors of a specific matrix. We also compute Gr\"obner bases and discuss to what extent our results carry over to the non-generic case.
The Parameter Continuation Theorem is the theoretical foundation for polynomial homotopy continuation, which is one of the main tools in computational algebraic geometry. In this note, we give a short proof using Gr\"obner bases. Our approach gives a method for computing discriminants.
Numerical nonlinear algebra is a computational paradigm that uses numerical analysis to study polynomial equations. Its origins were methods to solve systems of polynomial equations based on the classical theorem of B\'ezout. This was decisively linked to modern developments in algebraic geometry by the polyhedral homotopy algorithm of Huber and Sturmfels, which exploits the combinatorial structure of the equations and led to efficient software for solving polynomial equations. Subsequent growth of numerical nonlinear algebra continues to be informed by algebraic geometry and its applications. These include new approaches to solving, algorithms for studying positive-dimensional varieties, certification, and a range of applications both within mathematics and from other disciplines. With new implementations, numerical nonlinear algebra is now a fundamental computational tool for algebraic geometry and its applications. We survey some of these innovations and some recent applications.
The essential variety is an algebraic subvariety of dimension $5$ in real projective space $\mathbb R\mathrm P^{8}$ which encodes the relative pose of two calibrated pinhole cameras. The $5$-point algorithm in computer vision computes the real points in the intersection of the essential variety with a linear space of codimension $5$. The degree of the essential variety is $10$, so this intersection consists of 10 complex points in general. We compute the expected number of real intersection points when the linear space is random. We focus on two probability distributions for linear spaces. The first distribution is invariant under the action of the orthogonal group $\mathrm{O}(9)$ acting on linear spaces in $\mathbb R\mathrm P^{8}$. In this case, the expected number of real intersection points is equal to $4$. The second distribution is motivated from computer vision and is defined by choosing 5 point correspondences in the image planes $\mathbb R\mathrm P^2\times \mathbb R\mathrm P^2$ uniformly at random. A Monte Carlo computation suggests that with high probability the expected value lies in the interval $(3.95 - 0.05,\ 3.95 + 0.05)$.
We present an algebraic study of line correspondences for pinhole cameras, in contrast to the thoroughly studied point correspondences. We define the line multiview variety as the Zariski closure of the image of the map projecting lines in 3-space to tuples of image lines in 2-space. We prove that in the case of generic camera matrices the line multiview variety is a determinantal variety and we provide a complete set-theoretic description for any camera arrangement. We investigate basic properties of this variety such as dimension, smoothness, and multidegree. Finally, we give experimental results for the Euclidean distance degree and robustness under noise for the triangulation of lines.
In this paper we study circles tangent to conics. We show there are generically $184$ complex circles tangent to three conics in the plane and we characterize the real discriminant of the corresponding polynomial system. We give an explicit example of $3$ conics with $136$ real circles tangent to them. We conjecture that 136 is the maximal number of real circles. Furthermore, we implement a hill-climbing algorithm to find instances of conics with many real circles, and we introduce a machine learning model that, given three real conics, predicts the number of circles tangent to these three conics.
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.