arXiv subjects

Nayda Farnsworth

Publications and source records attributed to Nayda Farnsworth.

4 recordsLinked to original sources

arXiv2026

Numerical Elimination: Computing Complements of Real Hypersurfaces Using Pseudo-Witness Sets

Many hypersurfaces in algebraic geometry, such as discriminants, arise as the projection of another variety. The real complement of such a hypersurface decomposes into connected components. In this paper, we propose a new method for computing these components. Existing methods require the explicit equation of the hypersurface as input. However, computing this equation by elimination can be computationally demanding or even infeasible. Our approach instead derives from univariate interpolation by computing the intersection of the hypersurface with a line. Such an intersection may be computed using so-called pseudo-witness sets without computing a defining equation for the hypersurface. We implement our approach in a forthcoming Julia package and demonstrate, on several examples, that the resulting algorithm accurately recovers all components of the real complement of the hypersurface.

math.AG↗

arXiv2026

On Hales-Jewett and Related Numbers

We investigate the Hales-Jewett numbers and some variants of them, both computationally and via enumerative and probabilistic arguments. In particular, we give the improved lower bound formula on the combinatorial-geometric variant of the Hales-Jewett numbers. {As a consequence of one of the variants investigated, we show that the Milton Bradley game Connect Four played with 2 players on a 5-dimensional hypercube of any side length can end in a draw.

math.CO↗

arXiv2026

A Note on Conjectures of Gullerud, Johnson, and Mbirika

In 2023, Gullerud, Johnson, and Mbirika presented results on their study of certain tridiagonal real symmetric matrices. As part of their work, they studied the roots to nonhomogeneous equations related to characteristic polynomials of adjacency matrices for path graphs. They showed that a subset of these polynomials give a Fibonacci number when evaluated at the imaginary unit, leading them to make several intriguing conjectures. In this work, we further explore their conjectures regarding the distribution of roots. We make partial progress towards establishing two conjectures, identify an infinite class of polynomials for which a third is false, and give evidence against a fourth.

math.CO↗

arXiv2026

Can Neural Networks Learn Small Algebraic Worlds? An Investigation Into the Group-theoretic Structures Learned By Narrow Models Trained To Predict Group Operations

While a real-world research program in mathematics may be guided by a motivating question, the process of mathematical discovery is typically open-ended. Ideally, exploration needed to answer the original question will reveal new structures, patterns, and insights that are valuable in their own right. This contrasts with the exam-style paradigm in which the machine learning community typically applies AI to math. To maximize progress in mathematics using AI, we will need to go beyond simple question answering. With this in mind, we explore the extent to which narrow models trained to solve a fixed mathematical task learn broader mathematical structure that can be extracted by a researcher or other AI system. As a basic test case for this, we use the task of training a neural network to predict a group operation (for example, performing modular arithmetic or composition of permutations). We describe a suite of tests designed to assess whether the model captures significant group-theoretic notions such as the identity element, commutativity, or subgroups. Through extensive experimentation we find evidence that models learn representations capable of capturing abstract algebraic properties. For example, we find hints that models capture the commutativity of modular arithmetic. We are also able to train linear classifiers that reliably distinguish between elements of certain subgroups (even though no labels for these subgroups are included in the data). On the other hand, we are unable to extract notions such as the concept of the identity element. Together, our results suggest that in some cases the representations of even small neural networks can be used to distill interesting abstract structure from new mathematical objects.

cs.LG↗