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Anton Leykin

Publications and source records attributed to Anton Leykin.

At least 37 records · Page 2Linked to original sources
arXiv2020

Numerical Schubert Calculus via the Littlewood-Richardson Homotopy Algorithm

We develop the Littlewood-Richardson homotopy algorithm, which uses numerical continuation to compute solutions to Schubert problems on Grassmannians and is based on the geometric Littlewood-Richardson rule. One key ingredient of this algorithm is our new optimal formulation of Schubert problems in local Stiefel coordinates as systems of equations. Our implementation can solve problem instances with tens of thousands of solutions.

math.AG↗
arXiv2020

Noetherian operators and primary decomposition

Noetherian operators are differential operators that encode primary components of a polynomial ideal. We develop a framework, as well as algorithms, for computing Noetherian operators with local dual spaces, both symbolically and numerically. For a primary ideal, such operators provide an alternative representation to one given by a set of generators. This description fits well with numerical algebraic geometry, taking a step toward the goal of numerical primary decomposition.

math.AG↗
arXiv2020

A numerical toolkit for multiprojective varieties

A numerical description of an algebraic subvariety of projective space is given by a general linear section, called a witness set. For a subvariety of a product of projective spaces (a multiprojective variety), the corresponding numerical description is given by a witness collection, whose structure is more involved. We build on recent work to develop a toolkit for the numerical manipulation of multiprojective varieties that operates on witness collections, and use this toolkit in an algorithm for numerical irreducible decomposition of multiprojective varieties. The toolkit and decomposition algorithm are illustrated throughout in a series of examples.

math.AG↗
arXiv2020

PL${}_{1}$P -- Point-line Minimal Problems under Partial Visibility in Three Views

We present a complete classification of minimal problems for generic arrangements of points and lines in space observed partially by three calibrated perspective cameras when each line is incident to at most one point. This is a large class of interesting minimal problems that allows missing observations in images due to occlusions and missed detections. There is an infinite number of such minimal problems; however, we show that they can be reduced to 140616 equivalence classes by removing superfluous features and relabeling the cameras. We also introduce camera-minimal problems, which are practical for designing minimal solvers, and show how to pick a simplest camera-minimal problem for each minimal problem. This simplification results in 74575 equivalence classes. Only 76 of these were known; the rest are new. In order to identify problems that have potential for practical solving of image matching and 3D reconstruction, we present several smaller natural subfamilies of camera-minimal problems as well as compute solution counts for all camera-minimal problems which have less than 300 solutions for generic data.

cs.CV↗
arXiv2019

PLMP -- Point-Line Minimal Problems in Complete Multi-View Visibility

We present a complete classification of all minimal problems for generic arrangements of points and lines completely observed by calibrated perspective cameras. We show that there are only 30 minimal problems in total, no problems exist for more than 6 cameras, for more than 5 points, and for more than 6 lines. We present a sequence of tests for detecting minimality starting with counting degrees of freedom and ending with full symbolic and numeric verification of representative examples. For all minimal problems discovered, we present their algebraic degrees, i.e. the number of solutions, which measure their intrinsic difficulty. It shows how exactly the difficulty of problems grows with the number of views. Importantly, several new minimal problems have small degrees that might be practical in image matching and 3D reconstruction.

cs.CV↗
arXiv2019

Effective certification of approximate solutions to systems of equations involving analytic functions

We develop algorithms for certifying an approximation to a nonsingular solution of a square system of equations built from univariate analytic functions. These algorithms are based on the existence of oracles for evaluating basic data about the input analytic functions. One approach for certification is based on alpha-theory while the other is based on the Krawczyk generalization of Newton's iteration. We show that the necessary oracles exist for D-finite functions and compare the two algorithmic approaches for this case using our software implementation in SageMath.

cs.SC↗
arXiv2019

Stillman's conjecture via generic initial ideals

Using recent work by Erman-Sam-Snowden, we show that finitely generated ideals in the ring of bounded-degree formal power series in infinitely many variables have finitely generated Gröbner bases relative to the graded reverse lexicographic order. We then combine this result with the first author's work on topological Noetherianity of polynomial functors to give an algorithmic proof of the following statement: ideals in polynomial rings generated by a fixed number of homogeneous polynomials of fixed degrees only have a finite number of possible generic initial ideals, independently of the number of variables that they involve and independently of the characteristic of the ground field. Our algorithm outputs not only a finite list of possible generic initial ideals, but also finite descriptions of the corresponding strata in the space of coefficients.

math.AC↗
arXiv2018

Monodromy Solver: Sequential and Parallel

We describe, study, and experiment with an algorithm for finding all solutions of systems of polynomial equations using homotopy continuation and monodromy. This algorithm follows a framework developed in previous work and can operate in the presence of a large number of failures of the homotopy continuation subroutine. We give special attention to parallelization and probabilistic analysis of a model adapted to parallelization and failures. Apart from theoretical results, we developed a simulator that allows us to run a large number of experiments without recomputing the outcomes of the continuation subroutine.

cs.SC↗
arXiv2018

Solving polynomial systems via homotopy continuation and monodromy

We study methods for finding the solution set of a generic system in a family of polynomial systems with parametric coefficients. We present a framework for describing monodromy based solvers in terms of decorated graphs. Under the theoretical assumption that monodromy actions are generated uniformly, we show that the expected number of homotopy paths tracked by an algorithm following this framework is linear in the number of solutions. We demonstrate that our software implementation is competitive with the existing state-of-the-art methods implemented in other software packages.

math.AG↗
arXiv2017

Equivariant Gröbner bases

Algorithmic computation in polynomial rings is a classical topic in mathematics. However, little attention has been given to the case of rings with an infinite number of variables until recently when theoretical efforts have made possible the development of effective routines. Ability to compute relies on finite generation up to symmetry for ideals invariant under a large group or monoid action, such as the permutations of the natural numbers. We summarize the current state of theory and applications for equivariant Gröbner bases, develop several algorithms to compute them, showcase our software implementation, and close with several open problems and computational challenges.

math.AC↗
arXiv2017

Beyond Polyhedral Homotopies

We present a new algorithmic framework which utilizes tropical geometry and homotopy continuation for solving systems of polynomial equations where some of the polynomials are generic elements in linear subspaces of the polynomial ring. This approach generalizes the polyhedral homotopies by Huber and Sturmfels.

math.AG↗
arXiv2017

Trace test

The trace test in numerical algebraic geometry verifies the completeness of a witness set of an irreducible variety in affine or projective space. We give a brief derivation of the trace test and then consider it for subvarieties of products of projective spaces using multihomogeneous witness sets. We show how a dimension reduction leads to a practical trace test in this case involving a curve in a low-dimensional affine space.

math.AG↗
arXiv2016

Determinantal representations of hyperbolic curves via polynomial homotopy continuation

A smooth curve in the real projective plane is hyperbolic if its ovals are maximally nested. By the Helton-Vinnikov Theorem, any such curve admits a definite symmetric determinantal representation. We use polynomial homotopy continuation to compute such representations numerically. Our method works by lifting paths from the space of hyperbolic polynomials to a branched cover in the space of pairs of symmetric matrices.

math.AG↗
arXiv2016

Hilbert series of symmetric ideals in infinite polynomial rings via formal languages

Let $R$ be the polynomial ring $K[x_{i,j}]$ where $1 \le i \le r$ and $j \in \mathbb{N}$, and let $I$ be an ideal of $R$ stable under the natural action of the infinite symmetric group $S_{\infty}$. Nagel--Römer recently defined a Hilbert series $H_I(s,t)$ of $I$ and proved that it is rational. We give a much shorter proof of this theorem using tools from the theory of formal languages and a simple algorithm that computes the series.

math.AC↗
arXiv2015

Survey on the $D$-module $f^s$

In this survey we discuss various aspects of the singularity invariants with differential origin derived from the $D$-module generated by $f^s$.

math.AG↗
arXiv2015

Noetherianity for infinite-dimensional toric varieties

We consider a large class of monomial maps respecting an action of the infinite symmetric group, and prove that the toric ideals arising as their kernels are finitely generated up to symmetry. Our class includes many important examples where Noetherianity was recently proved or conjectured. In particular, our results imply Hillar-Sullivant's Independent Set Theorem and settle several finiteness conjectures due to Aschenbrenner, Martin del Campo, Hillar, and Sullivant. We introduce a matching monoid and show that its monoid ring is Noetherian up to symmetry. Our approach is then to factorize a more general equivariant monomial map into two parts going through this monoid. The kernels of both parts are finitely generated up to symmetry: recent work by Yamaguchi-Ogawa-Takemura on the (generalized) Birkhoff model provides an explicit degree bound for the kernel of the first part, while for the second part the finiteness follows from the Noetherianity of the matching monoid ring.

math.AC↗
arXiv2015

Eliminating dual spaces

Macaulay dual spaces provide a local description of an affine scheme and give rise to computational machinery that is compatible with the methods of numerical algebraic geometry. We introduce eliminating dual spaces, use them for computing dual spaces of quotient ideals, and develop an algorithm for detection of embedded points on an algebraic curve.

math.AG↗