Anton Betten (Colorado State)
Abstract
Recently, the classifiation of cubic surfaces with 27 lines over small finite fields up to projective equivalence has received renewed interest. In [2], the cubic surfaces with 27 lines over the field with 13 elements were classified (there are 4 of them). This continues earlier work on classifiying surfaces by Hirschfeld (q ≤ 9) and Sadeh (q = 11). Several infinite families of cubic surfaces are known. The data obtained from computer classifications suggests that there are many more to be discovered.
There are two different methods that can be employed. In [3], cubic surfaces are classified using the classical theory around double sixes in pro- jective three spaces. In [4], the classical theory of Clebsch maps (or blow ups) of six general points in a plane is described. In this talk, we summarize these developments and we compare some of the results. We describe imple- mentations of these computer searches and classifications in C++ using the software package Orbiter [1]. Based on these computations, a new family of cubic surfaces was found and described in [3].
References
[1] Anton Betten. Classifying Discrete Objects with Orbiter. ACM Communications in Computer Algebra 01/2014; 47(3/4):183-186. DOI:10.1145/2576802.2576832
[2] Anton Betten, James W. P. Hirschfeld, Fatma Karaoglu. Classification of cubic surfaces with twenty-seven lines over the finite field of order thirteen. Eur. J. Math. 4 (2018), no. 1, 3750.
[3] Anton Betten, Fatma Karaoglu. Cubic surfaces over small finite fields. Submitted to Designs, Codes and Cryptography.
[4] Fatma Karaoglu. The cubic surfaces with twenty-seven lines over finite fields, Thesis, University of Sussex, 2018. Submitted.
Michael Burr (Clemson), Juan Xu (Beihang University), and Chee Yap (NYU)
Abstract
Homotopy continuation is a well-known approach in numerical root-finding. Traditional homotopy continuation algorithms are not guaranteed to track a single path since path jumping may occur. Recently, the authors have developed and implemented an adaptive precision and certified algorithm for homotopy continuation. This approach is based on well-isolated approximations to roots and uses interval arithmetic, the Graeffe-Pellet test, and subdivision to provide a certified approximation to the solution curve. Our algorithm has been implemented in c++ and experimental evidence illustrates its feasibility. In this talk, I will provide a theoretical overview of this algorithm and discuss its implementation. In particular, I will present some of the details of the implementation with a particular focus on the certification and adaptive precision. I will also show how the modularity of our design allows others to contribute to this approach and test additional features.
Tianran Chen (Auburn-Montgomery)
Abstract
The computation of intersection points of generic tropical hyper-surfaces is a fundamental problem in computational algebraic ge- ometry. An efficient algorithm for solving this problem will be a basic building block in many higher level algorithms for studying tropical va- rieties, computing mixed volume, enumerating mixed cells, constructing polyhedral homotopies, etc. libtropicon is a library for computing inter- section points of generic tropical hyper-surfaces that provides a unified framework where the several conceptually opposite approaches coexist and complement one another. In particular, great efficiency is achieve by the data cross-feeding of the “pivoting” and the “elimination” step — data by-product generated by the pivoting step is selectively saved to bootstrap the elimination step, and vice versa. The core algorithm is designed to be naturally parallel and highly scalable, and the implemen- tation directly supports multi-core architectures, computer clusters, and GPUs based on CUDA or ROCm/OpenCL technology. Many-core archi- tectures such as Intel Xeon Phi are also partially supported. This library also includes interface layers that allows it to be tightly integrated into the existing ecosystem of software in computational algebraic geometry.
Alicia Dickenstein (University of Buenos Aires), Mercedes Perez Millan (University of Buenos Aires), Anne Shiu (Texas A&M), and Xiaoxian Tang (Texas A&M)
Abstract
Many dynamical systems arising in applications exhibit multistationarity (two or more positive steady states), but it is often difficult to determine whether a given system is multistationary, and if so to identify a witness to multistationarity, that is, specific parameter values for which the system exhibits multiple steady states. In this talk we introduce a procedure to investigate multistationarity and to find a witness. In practice, the procedure is much less expensive than traditional quantifier elimination. Our method is based on two new sufficient conditions for multistationarity. First, when there are no boundary steady states and a positive steady-state parametrization exists, one can conclude multistationarity if a certain critical function changes sign. Particularly, if the steady states are defined by binomials, we have multistationarity if a certain critical function contains terms with different signs. Second, when the steady-state equations can be replaced by equivalent triangular-form equations, we have multistationarity if a positive degenerate steady state exists. We also investigate the mathematical structure of this critical function, and give conditions that guarantee that triangular-form equations exist by studying the specialization of Grobner bases.
Jaime Gutierrez (University of Cantabria, Santander) and Jorge Jimenez Urroz (Polytechnic University of Catalonia, Barcelona)
Abstract
The aim of the talk is producing new families of irreducible polynomials, generalizing previous results in the area. One example of our general result is that for a near-separated polynomial, i.e., polynomials of the form $F(x,y)=f_1(x)f_2(y)-f_2(x)f_1(y)$, then $F(x,y)+r$ is always irreducible for any constant $r$ different from zero. The proof of this result is based on somehow generalization of the well celebrated Eisenstein's Irreducibility Criterion:
{\bf Eisenstein Criterion}
Let $\phi(x,y)=(\phi_1(x,y),\phi_2(x,y))$ be an automorphism of $\K[x,y]$. Consider the polynomial $F(x,y)=\phi_1Q(\phi_1,\phi_2)+r(\phi_1)$ such that $r(0)\ne 0$, $Q(\phi_2,\phi_2)=\sum_{j=0}^dp_j(\phi_1)\phi_2^i$ verifies $\phi_1\nmid p_d(\phi_1)$ and $\gcd(p_0,p_1,...,p_d,r) = 1$. Then $F(x,y)$ is absolutely irreducible.
We also present the computational problem of how to recognize whether a given polynomial $F(x,y)$ is of the above form.
Jonathan Hauenstein (Notre Dame), Avinash Kulkarni (Simon Fraser), Emre C. Sertöz (Max-Planck, Leipzig), and Samantha Sherman (Notre Dame)
Abstract
A common topic in computational mathematics is the ability to generate certified
results using numerical computations. For systems of equations, one aims to produce certified
results regarding solutions from numerical approximations. For example, Smale's alpha theory
can be used to certify that a given point is in the quadratic convergence basin of some
solution. Recent work has extended this to certify results about the corresponding solution,
such as deciding reality of a solution. In this work, we utilize the theory of Newton-invariant
sets to certifiably determine the reality of projections of a solution. This approach is then
applied to certifiably count the number of real and totally real tritangent hyperplanes for curves
of genus 4 which used Bertini to numerically approximate solutions and alphaCertified to certify
the solutions.
Robert H. Lewis (Fordham University)
Abstract
The problem is to identify a movable object that is in some sense known if it is encountered later. Suppose we have a sensor, on a fixed radar station or a moving platform. We have an object, say object A, previously measured, with certain distinct identifiable points pi. We know the distances between these points. We later encounter a similar object B and want to know if it is A. We have a sensor that sends and receives electronic signals, and so we measure the distances ti from the sensor to the distinguished points on B.
We first consider the two-dimensional case. Assume there are three distinct points on A. We have our measured distances t1,t2,t3 and previously known distances be- tween the points on A, d1, d2, d3. We derive a polynomial system relating these quan- tities and show that it is easy to solve yielding a resultant that is the ”signature” for A. Its use will eliminate B if B is not A.
The generalization to three dimensions is immediate. We need a fourth point. The polynomial system contains many parameters, but we solve it. We then discuss generalizations, such as what to do if one of the distinguished points is hidden. In that case we need five points and the system is much more complex.
We compare solutions on Mathematica, Maple, and Fermat computer algebra systems.
Anton Leykin (Georgia Tech)
Abstract
We describe the design and relationships of several Macaulay2 packages that use numerical polynomial homotopy continuation as their engine.
Macaulay2 is a computer algebra system built around the classical symbolic computation tools such as Groebner bases. However, recent Macaulay2 versions include its own fast implementation of homotopy continuation, interfaces to external numerical algebraic geometry software (Bertini and PHCpack), and a unified data structures design that allows to use the internal and external capabilities interchangeably.
The resulting numerical and hybrid tools are of general interest to Macaulay2 users interested in computational experimentation.
Nida Obatake (Texas A&M)
Abstract
Chemical Reaction Network theory is an area of mathematics that analyzes the behaviors of chemical processes. A major problem in this area is stability in these networks. This talk focuses on bifurcations in a particular network, the fully distributive dual-site phosphorylation network. Experimental results suggest that this network does not exhibit bifurcations, but as far as we know, there are no theoretical results to support this conjecture. In this work we examine the capacity for Hopf bifurcations, by analyzing the steady state locus of the ODE system obtained from the network. To reduce the number of variables, we compute a parameterization of the steady state locus. We use Maple to compute the corresponding Hurwitz matrix and its minors, so that we may apply a generalization of the Routh-Hurwitz criterion for Hopf bifurcations. Using SAGE, we examine the Newton polytope to understand the signs that the Hurwitz determinants take. We show for the first time that the relevant Hurwitz determinants change sign, and discuss the implications for bifurcations and oscillations in the network. Joint work with Anne Shiu and Xiaoxian Tang.
Ethan Petersen (Rose-Hulman), Nora Youngs (Colby College), Ryan Kruse (Central College), Dane Miyata (Willamette University), Rebecca Garcia (Sam Houston State), and Luis David Garcia Puente (Sam Houston State)
Abstract
A major area in neuroscience research is the study of how the brain processes spatial information. Neurons in the brain represent external stimuli via neural codes. These codes often arise from stereotyped stimulus-response maps, associating to each neuron a convex receptive field. An important problem consists in determining what stimulus space features can be extracted directly from a neural code. The neural ideal is an algebraic object that encodes the full combinatorial data of a neural code. This ideal can be expressed in a canonical form that directly translates to a minimal description of the receptive field structure intrinsic to the code. In here, we describe a SageMath package that contains several algorithms related to the canonical form of a neural ideal.
Margaret Regan (Notre Dame), Jonathan Hauenstein (Notre Dame), and Danielle Brake (Wisconsin - Eau Claire)
Abstract
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 smooth real surface. Several examples are used to demonstrate this new software.
Jose Rodriguez (University of Chicago)
Abstract
Studying singularities of algebraic varieties is of great interest in applied and computational algebraic geometry. For example, in applications the singular locus is important when finding the closest point to an algebraic variety. In computational algebraic geometry, the singular locus can lead to bottlenecks of an algorithm. One way to understand a singular algebraic variety is by stratifying it into locally closed subvarieties called Whitney strata. Then, for each stratum one considers the local information at a point.
Macpherson defined Chern-Schwartz-Macpherson (CSM) classes by introducing the (local) Euler obstruction function, which is an integer valued function on the variety that is constant on each stratum. By understanding the Euler obstruction function, one gains insights about the singular algebraic variety. In this talk, I will discuss how to use maximum likelihood degrees to compute the Euler obstruction at a point. The key idea is that we do not need to know the Whitney stratification to run our algorithm. Macaulay2 and Bertini are used in the implementation.
Sascha Timme (TU-Berlin)
Abstract
We present the Julia package HomotopyContinuation.jl (www.JuliaHomotopyContinuation.org) for solving polynomial systems by numerical homotopy continuation.
Julia is a high-level, high-performance dynamic programming language for numerical computing and the goal of our package is to provide a fast polynomial systems solver for the Julia ecosystem while at the same time keeping an intuitive user interface.
A key aspect of the package is its modular design. This enables the user to adapt the solver to specific applications as well as to extend its capabilities in relatively few steps and with a reasonable amount of effort. For instance, it is straight-forward to implement custom homotopies and custom path tracking algorithms; all within Julia and without any loss in performance compared to the capabilities provided by default. Since the input is defined in Julia one can make use of the extensive mathematical function library as well as the growing package ecosystem to construct the input.
In this talk, we give an introduction to HomotopyContinuation.jl, outline its current capabilities and discuss future perspectives. We also take a closer look at the design of the software and show how its design in the combination with some of the distinct features of Julia enables us to achieve our outlined goal.
Furthermore, we show some examples which demonstrate the advantages of the modularity of our package.
This is joint work with Paul Breiding (MPI MiS Leipzig).
Mark C. Wilson (U. Auckland)
Abstract
Work by Robin Pemantle, the speaker, and several others has developed a substantial theory allowing asymptotic coefficient extraction from sufficiently nice (e.g. rational) multivariate generating functions. It involves algebraic-geometric computations at points of the variety given by the vanishing set of the denominator. A package to perform the computations, written by Alex Raichev, is now included in the core Sage distribution. It works well on cases where the relevant computations only require global factorization. However the general theory, motivated by important examples, involves local computations, and these are not implemented yet. I will describe the functionality of this package and hope to obtain audience feedback on improving it.
Dan Bates (Colorado State), Danielle Brake (Wisconsin - Eau Claire), and Matt Niemerg
Abstract
Numerical algebraic geometry provides tools for approximating solutions of
polynomial systems. One such tool is the parameter homotopy, which can be an extremely efficient method
to solve numerous polynomial systems that differ only in coefficients, not monomials. This technique is frequently
used for solving a parameterized family of polynomial systems at multiple parameter values.
This article describes Paramotopy, a
parallel, optimized implementation of this technique, making use of the
Bertini software package. The novel features of this implementation
include allowing for the simultaneous solutions of arbitrary polynomial systems in a parameterized family on
an automatically generated or manually provided mesh in the parameter space of coefficients,
front ends and back ends that are easily specialized to particular classes of problems, and
adaptive techniques for solving polynomial systems near singular points in the parameter space.