Applied and Computational Mathematics and Statistics
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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:
 20200421
 Record Visibility:
 Public

2
Doctoral Dissertation

 Creator(s):
 Jonathan Hauenstein, Margaret Regan
 Description:
Polynomials which arise via elimination can be difficult to compute explicitly. By using a pseudowitness 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:
 20200327
 Record Visibility:
 Public

4
Doctoral Dissertation

5
Doctoral Dissertation

 Creator(s):
 Jonathan Hauenstein, Martin Helmer
 Description:
Alt’s problem, formulated in 1923, is to count the number of fourbar 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:
 20200303
 Record Visibility:
 Public

7
Dataset
 Creator(s):
 Jonathan Hauenstein
 Description:
Many algorithms for determining properties of real algebraic or semialgebraic sets rely upon the ability to compute smooth points. Existing methods to compute smooth points on semialgebraic sets use symbolic quantifier elimination tools. In this paper, we present a simple algorithm based on computing the critical points of some wellchosen 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:
 20200118
 Record Visibility:
 Public

8
Doctoral Dissertation

9
Doctoral Dissertation

10
Doctoral Dissertation

 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 dimensionbydimension 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:
 20190426
 Record Visibility:
 Public

12
Doctoral Dissertation