133 citations · 554 across the 117 of their papers we have counts for
84 papers · 1 filter
Some problems at the interface of approximation theory and algebraic geometry
Frank Sottile
We sketch an assortment of problems that were posed -- and not yet solved -- during problem sessions at the conference ``Approximation Theory and Numerical Analysis meet Algebra, G…
Numerical Nonlinear Algebra
Daniel J. Bates, Paul Breiding, Tianran Chen +3
Numerical nonlinear algebra is a computational paradigm that uses numerical analysis to study polynomial equations. Its origins were methods to solve systems of polynomial equation…
Real solutions to systems of polynomial equations in Macaulay2
Jordy Lopez Garcia, Kelly Maluccio, Frank Sottile +1
The Macaulay2 package RealRoots provides symbolic methods to study real solutions to systems of polynomial equations. It updates and expands an earlier package developed by Grayson…
Galois Groups in Enumerative Geometry and Applications
Frank Sottile, Thomas Yahl
As Jordan observed in 1870, just as univariate polynomials have Galois groups, so do problems in enumerative geometry. Despite this pedigree, the study of Galois groups in enumerat…
Euclidean distance degree and mixed volume
Paul Breiding, Frank Sottile, James Woodcock
We initiate a study of the Euclidean Distance Degree in the context of sparse polynomials. Specifically, we consider a hypersurface f=0 defined by a polynomial f that is general gi…
Numerical homotopies from Khovanskii bases
Michael Burr, Frank Sottile, Elise Walker
We present numerical homotopy continuation algorithms for solving systems of equations on a variety in the presence of a finite Khovanskii basis. These take advantage of Anderson's…
Decomposable sparse polynomial systems
Taylor Brysiewicz, Jose Israel Rodriguez, Frank Sottile +1
The Macaulay2 package DecomposableSparseSystems implements methods for studying and numerically solving decomposable sparse polynomial systems. We describe the structure of decompo…
Nontrivial linear projections of the Grassmannian Gr_3(C^6)
Yanhe Huang, George Petroulakis, Frank Sottile +1
A typical linear projection of the Grassmannian in its Plucker embedding is injective, unless its image is a projective space. A notable exception are self-adjoint linear projectio…
General witness sets for numerical algebraic geometry
Frank Sottile
Numerical algebraic geometry has a close relationship to intersection theory from algebraic geometry. We deepen this relationship, explaining how rational or algebraic equivalence…
Solving Decomposable Sparse Systems
Taylor Brysiewicz, Jose Israel Rodriguez, Frank Sottile +1
Amendola et al. proposed a method for solving systems of polynomial equations lying in a family which exploits a recursive decomposition into smaller systems. A family of systems a…
On the fundamental group of open Richardson varieties
Changzheng Li, Frank Sottile, Chi Zhang
We compute the fundamental group of an open Richardson variety in the manifold of complete flags that corresponds to a partial flag manifold. Rietsch showed that these log Calabi-Y…
Galois Groups of Composed Schubert Problems
Frank Sottile, Robert Williams, Li Ying
Two Schubert problems on possibly different Grassmannians may be composed to obtain a Schubert problem on a larger Grassmannian whose number of solutions is the product of the numb…
A numerical toolkit for multiprojective varieties
Jonathan D. Hauenstein, Anton Leykin, Jose Israel Rodriguez +1
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 space…
Classification of Schubert Galois groups in Gr(4,9)
Abraham Martin del Campo, Frank Sottile, Robert Williams
We classify Schubert problems in the Grassmannian of 4-planes in 9-dimensional space by their Galois groups. Of the 31,806 essential Schubert problems in this Grassmannian, there a…
Certification for Polynomial Systems via Square Subsystems
Timothy Duff, Nickolas Hein, Frank Sottile
We consider numerical certification of approximate solutions to a system of polynomial equations with more equations than unknowns by first certifying solutions to a square subsyst…
Irrational Toric Varieties
Ata Firat Pir
Classical toric varieties are among the simplest objects in algebraic geometry. They arise in an elementary fashion as varieties parametrized by monomials whose exponents are a fin…
A geometric proof of an equivariant Pieri rule for flag manifolds
Changzheng Li, Vijay Ravikumar, Frank Sottile +1
We use geometry to give a short proof of an equivariant Pieri rule in the classical flag manifold. This rule is due to Robinson, who gave an algebraic proof.
Irrational toric varieties and secondary polytopes
Ata Pir, Frank Sottile
The space of torus translations and degenerations of a projective toric variety forms a toric variety associated to the secondary fan of the integer points in the polytope correspo…
Numerical Schubert Calculus via the Littlewood-Richardson Homotopy Algorithm
Anton Leykin, Abraham Martin del Campo, Frank Sottile +2
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…
Ibadan Lectures on Toric Varieties
Frank Sottile
Toric varieties are perhaps the most accessible class of algebraic varieties. They often arise as varieties parameterized by monomials, and their structure may be completely unders…
Convex Hull of Two Circles in R^3
Evan D. Nash, Ata Firat Pir, Frank Sottile +1
We describe convex hulls of the simplest compact space curves, reducible quartics consisting of two circles. When the circles do not meet in complex projective space, their algebra…
Injectivity of generalized Wronski maps
Yanhe Huang, Frank Sottile, Igor Zelenko
We study linear projections on Pluecker space whose restriction to the Grassmannian is a non-trivial branched cover. When an automorphism of the Grassmannian preserves the fibers,…
Trace test
Anton Leykin, Jose Israel Rodriguez, Frank Sottile
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…
Real Algebraic Geometry for Geometric Constraints
Frank Sottile
Real algebraic geometry adapts the methods and ideas from (complex) algebraic geometry to study the real solutions to systems of polynomial equations and polynomial inequalities. A…