Datasets & Related Materials
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List of files deposited in CurateND that match your search criteria
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1
Dataset
- Creator(s):
- Joseph Cummings, Jonathan Hauenstein
- Description:
This is Macaulay2 code that will compute the Macaulay dual space at 0 of a multi-graded ideal up to a specified multi-degree.
- Date Created:
- 2023-02-03
- Record Visibility:
- Public
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- Creator(s):
- Jonathan Hauenstein, Samantha Sherman
- Description:
Synthesis problems for linkages in kinematics often yield large structured parameterized polynomial systems which generically have far fewer solutions than traditional upper bounds would suggest. This paper describes statistical models for estimating the generic number of solutions of such parameterized polynomial systems. The new approach extends previous work on success ratios of parameter homotopies to using monodromy loops as well as the addition of a trace test that provides a stopping…
- Date Created:
- 2020-04-21
- Record Visibility:
- Public
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- Creator(s):
- Jonathan Hauenstein, Margaret Regan
- Description:
Polynomials which arise via elimination can be difficult to compute explicitly. By using a pseudo-witness set, we develop an algorithm to explicitly compute the restriction of a polynomial to a given line. The resulting polynomial can then be used to evaluate the original polynomial and directional derivatives along the line at any point on the given line. Several examples are used to demonstrate this new algorithm including examples of computing the critical points of the discriminant locu…
- Date Created:
- 2020-03-27
- Record Visibility:
- Public
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- Creator(s):
- Jonathan Hauenstein, Martin Helmer
- Description:
Alt’s problem, formulated in 1923, is to count the number of four-bar linkages whose coupler curve interpolates nine general points in the plane. This problem can be phrased as counting the number of solutions to a system of polynomial equations which was first solved numerically using homotopy continuation by Wampler, Morgan, and Sommese in 1992. Since there is still not a proof that all solutions were obtained, we consider upper bounds for Alt’s problem by counting the number of sol…
- Date Created:
- 2020-03-03
- Record Visibility:
- Public
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5
Dataset
- Creator(s):
- Jonathan Hauenstein
- Description:
Many algorithms for determining properties of real algebraic or semi-algebraic sets rely upon the ability to compute smooth points. Existing methods to compute smooth points on semi-algebraic sets use symbolic quantifier elimination tools. In this paper, we present a simple algorithm based on computing the critical points of some well-chosen function that guarantees the computation of smooth points in each connected compact component of a real (semi)-algebraic set. Our technique is intuitive …
- Date Created:
- 2020-01-18
- Record Visibility:
- Public
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- Creator(s):
- Dan Bates, David Eklund, Jonathan Hauenstein, Chris Peterson
- Description:
A fundamental problem in algebraic geometry is to decompose the solution set of a given polynomial system. A numerical description of this solution set is called a numerical irreducible decomposition and currently all standard algorithms use a sequence of homotopies forming a dimension-by-dimension approach. In this article, we pair a classical result to compute a smooth point on every irreducible component in every dimension using a single homotopy together with the theory of isosingular s…
- Date Created:
- 2019-04-26
- Record Visibility:
- Public
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- Creator(s):
- Jonathan Hauenstein, Jose Rodriguez
- Description:
In the field of numerical algebraic geometry, positive-dimensional solution sets of systems of polynomial equations are described by witness sets. In this paper, we define multiprojective witness sets which encode the multidegree information of an irreducible multiprojective variety. Our main results generalize the regeneration solving procedure, a trace test, and numerical irreducible decomposition to the multiprojective case. Examples are included to demonstrate this new approach.
- Date Created:
- 2018-12-29
- Record Visibility:
- Public
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- Creator(s):
- Jonathan Hauenstein, Alan Liddell, Sanesha McPherson, Yi Zhang
- Description:
Standard interior point methods in semidefinite programming can be viewed as tracking a solution path for a homotopy defined by a system of bilinear equations. By considering this in the context of numerical algebraic geometry, we employ numerical algebraic geometric techniques such as adaptive precision path tracking, endgames, and projective space to accurately solve semidefinite programs. We develop feasibility tests for both primal and dual problems which can distinguish between the fou…
- Date Created:
- 2018-04-10
- Record Visibility:
- Public
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- Creator(s):
- Jonathan Hauenstein
- Description:
A common problem when analyzing models, such as a mathematical modeling of a biological process, is to determine if the unknown parameters of the model can be determined from given input-output data. Identifiable models are models such that the unknown parameters can be determined to have a finite number of values given input-output data. The total number of such values over the complex numbers is called the identifiability degree of the model. Unidentifiable models are models such that th…
- Date Created:
- 2018-03-09
- Record Visibility:
- Public
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10
Dataset
- Creator(s):
- Samantha Sherman, Jonathan Hauenstein
- Description:
Computational tools in numerical algebraic geometry can be used to numerically approximate solutions to a system of polynomial equations. If the system is well-constrained (i.e., square), Newton’s method is locally quadratically convergent near each nonsingular solution. In such cases, Smale’s alpha theory can be used to certify that a given point is in the quadratic convergence basin of some solution. This was extended to certifiably determine the reality of the corresponding sol…
- Date Created:
- 2018-03-08
- Record Visibility:
- Public
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- Creator(s):
- Margaret Regan, Jonathan Hauenstein
- Description:
A common computational problem is to compute topological information about a real surface defined by a system of polynomial equations. Our software, called polyTop, leverages numerical algebraic geometry computations from Bertini and Bertini_real with topological computations in javaPlex to compute the Euler characteristic, genus, Betti numbers, and generators of the fundamental group of a real surface. Several examples are used to demonstrate this new software.
- Date Created:
- 2018-03-02
- Record Visibility:
- Public
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- Creator(s):
- Margaret Regan, Jonathan Hauenstein
- Description:
Three key aspects of applying homotopy continuation to parameterized systems of polynomial equations are investigated. First, for parameterized systems which are homogenized with solutions in projective space, we investigate options for selecting the affine patch where computations are performed. Second, for parameterized systems which are overdetermined, we investigate options for randomizing the system for improving the numerically stability of the computations. Finally, since one is typica…
- Date Created:
- 2017-07-06
- Record Visibility:
- Public