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Frank Sottile

Publications and source records attributed to Frank Sottile.

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

Certification for Polynomial Systems via Square Subsystems

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 subsystem. We give several approaches that certifiably select which are solutions to the original overdetermined system. These approaches each use different additional information for this certification, such as liaison, Newton-Okounkov bodies, or intersection theory. They may be used to certify individual solutions, reject nonsolutions, or certify that we have found all solutions.

math.AG↗
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

Galois Groups of Composed Schubert Problems

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 numbers of the original problems. This generalizes a construction discovered while classifying Schubert problems on the Grassmannian of 4-planes in C^9 with imprimitive Galois groups. We give an algebraic proof of the product formula. In a number of cases, we show that the Galois group of the composed Schubert problem is a subgroup of a wreath product of the Galois groups of the original problems, and is therefore imprimitive. We also present evidence for a conjecture that all composed Schubert problems have imprimitive Galois groups.

math.AG↗
arXiv2020

Decomposable sparse polynomial systems

The Macaulay2 package DecomposableSparseSystems implements methods for studying and numerically solving decomposable sparse polynomial systems. We describe the structure of decomposable sparse systems and explain how the methods in this package may be used to exploit this structure, with examples.

math.AG↗
arXiv2020

On the fundamental group of open Richardson varieties

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-Yau varieties underlie a Landau-Ginzburg mirror for the Langlands dual partial flag manifold, and our computation verifies a prediction of Hori for this mirror. It is log Calabi-Yau as it isomorphic to the complement of the Knutson-Lam-Speyer anti-canonical divisor for the partial flag manifold. We also determine explicit defining equations for this divisor.

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arXiv2020

General witness sets for numerical algebraic geometry

Numerical algebraic geometry has a close relationship to intersection theory from algebraic geometry. We deepen this relationship, explaining how rational or algebraic equivalence gives a homotopy. We present a general notion of witness set for subvarieties of a smooth complete complex algebraic variety using ideas from intersection theory. Under appropriate assumptions, general witness sets enable numerical algorithms such as sampling and membership. These assumptions hold for products of flag manifolds. We introduce Schubert witness sets, which provide general witness sets for Grassmannians and flag manifolds.

math.AG↗
arXiv2020

Generic properties of dispersion relations for discrete periodic operators

An old problem in mathematical physics deals with the structure of the dispersion relation of the Schrödinger operator $-Δ+V(x)$ in $R^n$ with periodic potential near the edges of the spectrum. A well known conjecture says that generically (with respect to perturbations of the periodic potential) the extrema are attained by a single branch of the dispersion relation, are isolated, and have non-degenerate Hessian (i.e., dispersion relations are graphs of Morse functions). The important notion of effective masses in solid state physics, as well as Liouville property, Green's function asymptotics, etc. hinge upon this property. The progress in proving this conjecture has been slow. It is natural to try to look at discrete problems, where the dispersion relation is (in appropriate coordinates) an algebraic, rather than analytic, variety. Such models are often used for computation in solid state physics (the tight binding model). Alas, counterexamples exist in some discrete situations. We start establishing the following dichotomy: the non-degeneracy of extrema either fails or holds in the complement of a proper algebraic subset of the parameters. The known counterexample has only two free parameters. This might be too tight for genericity to hold. We consider the maximal $Z^2$-periodic two-atomic nearest-cell interaction graph, with nine edges per unit cell and the discrete "Laplace-Beltrami" operator on it. We then use methods from computational and combinatorial algebraic geometry to prove the genericity conjecture for this graph. We show three different approaches to the genericity, which might be suitable in various situations. It is also proven in this case that adding more parameters does not destroy the genericity result. We list all "bad" periodic subgraphs of the one we consider and discover that in all these cases genericity fails for "trivial" reasons only.

math-ph↗
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

Nontrivial linear projections of the Grassmannian Gr_3(C^6)

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 projections, which have even degree. We consider linear projections of Gr_3(C^6) with low-dimensional centers of projection. When the center has dimension less than five, we show that the projection has degree 1. When the center has dimension five and the projection has degree greater than 1, we show that it is self-adjoint.

math.AG↗
arXiv2020

Bivariate Semialgebraic Splines

Semialgebraic splines are bivariate splines over meshes whose edges are arcs of algebraic curves. They were first considered by Wang, Chui, and Stiller. We compute the dimension of the space of semialgebraic splines in two extreme cases. If the polynomials defining the edges span a three-dimensional space of polynomials, then we compute the dimensions from the dimensions for a corresponding rectilinear mesh. If the mesh is sufficiently generic, we give a formula for the dimension of the spline space valid in large degree and bound how large the degree must be for the formula to hold. We also study the dimension of the spline space in examples which do not satisfy either extreme. The results are derived using commutative and homological algebra.

math.AC↗
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↗
arXiv2017

Convex Hull of Two Circles in R^3

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 algebraic boundary contains an irrational ruled surface of degree eight whose ruling forms a genus one curve. We classify which curves arise, classify the face lattices of the convex hulls, and determine which are spectrahedra. We also discuss an approach to these convex hulls using projective duality.

math.AG↗
arXiv2017

Injectivity of generalized Wronski maps

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, we show that the Grassmannian is necessarily of m-dimensional linear subspaces in a symplectic vector space of dimension 2m, and the linear map is the Lagrangian involution. The Wronski map for a self-adjoint linear differential operator and pole placement map for symmetric linear systems are natural examples.

math.AG↗
arXiv2016

Real Algebraic Geometry for Geometric Constraints

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. As it is the real solutions to such systems modeling geometric constraints that are physically meaningful, real algebraic geometry is a core mathematical input for geometric constraint systems.

math.AG↗
arXiv2016

Two Murnaghan-Nakayama rules in Schubert calculus

The Murnaghan-Nakayama rule expresses the product of a Schur function with a Newton power sum in the basis of Schur functions. We establish a version of the Murnaghan-Nakayama rule for Schubert polynomials and a version for the quantum cohomology ring of the Grassmannian. These rules compute all intersections of Schubert cycles with tautological classes coming from the Chern character.

math.CO↗
arXiv2016

Numerical computation of Galois groups

The Galois/monodromy group of a family of geometric problems or equations is a subtle invariant that encodes the structure of the solutions. Computing monodromy permutations using numerical algebraic geometry gives information about the group, but can only determine it when it is the full symmetric group. We give numerical methods to compute the Galois group and study it when it is not the full symmetric group. One algorithm computes generators while the other gives information on its structure as a permutation group. We illustrate these algorithms with examples using a Macaulay2 package we are developing that relies upon Bertini to perform monodromy computations.

math.AG↗
arXiv2016

Semialgebraic Splines

Semialgebraic splines are functions that are piecewise polynomial with respect to a cell decomposition into sets defined by polynomial inequalities. We study bivariate semialgebraic splines, formulating spaces of semialgebraic splines in terms of graded modules. We compute the dimension of the space of splines with large degree in two extreme cases when the cell decomposition has a single interior vertex. First, when the forms defining the edges span a two-dimensional space of forms of degree n---then the curves they define meet in n^2 points in the complex projective plane. In the other extreme, the curves have distinct slopes at the vertex and do not simultaneously vanish at any other point. We also study examples of the Hilbert function and polynomial in cases of a single vertex where the curves do not satisfy either of these extremes.

math.AC↗