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Nigel Boston

Publications and source records attributed to Nigel Boston.

At least 19 recordsLinked to original sources
arXiv2020

Heuristics for $2$-class Towers of Cyclic Cubic Fields

We consider the Galois group $G_2(K)$ of the maximal unramified $2$-extension of $K$ where $K/\mathbb{Q}$ is cyclic of degree $3$. We also consider the group $G^+_2(K)$ where ramification is allowed at infinity. In the spirit of the Cohen-Lenstra heuristics, we identify certain types of pro-$2$ group as the natural spaces where $G_2(K)$ and $G^+_2(K)$ live when the $2$-class group of $K$ is $2$-generated. While we do not have a theoretical scheme for assigning probabilities, we present data and make some observations and conjectures about the distribution of such groups.

math.NT↗
arXiv2020

Exploiting Symmetry in the Power Flow Equations Using Monodromy

We propose solving the power flow equations using monodromy. We prove the variety under consideration decomposes into trivial and nontrivial subvarieties and that the nontrivial subvariety is irreducible. We also show various symmetries in the solutions. We finish by giving numerical results comparing monodromy against polyhedral and total degree homotopy methods and giving an example of a network where we can find all solutions to the power flow equation using monodromy where other homotopy techniques fail. This work gives hope that finding all solutions to the power flow equations for networks of realistic size is possible.

math.AG↗
arXiv2020

The Distribution of the Number of Real Solutions to the Power Flow Equations

In this paper we study the distributions of the number of real solutions to the power flow equations over varying electrical parameters. We introduce a new monodromy and parameter homotopy continuation method for quickly finding all solutions to the power flow equations. We apply this method to find distributions of the number of real solutions to the power flow equations and compare these distributions to those of random polynomials. It is observed that while the power flow equations tend to admit many fewer real-valued solutions than a bound on the total number of complex solutions, for low levels of load they tend to admit many more than a corresponding random polynomial. We show that for cycle graphs the number of real solutions can achieve the maximum bound for specific parameter values and for complete graphs with four or more vertices there are susceptance values that give infinitely many real solutions.

math.AG↗
arXiv2018

Heuristics for p-class towers of real quadratic fields

Let $p$ be an odd prime. For a number field $K$, we let $K_\infty$ be the maximal unramified pro-$p$ extension of $K$; we call the group $\mathrm{Gal}(K_\infty/K)$ the $p$-class tower group of $K$. In a previous work, as a non-abelian generalization of the work of Cohen and Lenstra on ideal class groups, we studied how likely it is that a given finite $p$-group occurs as the $p$-class tower group of an imaginary quadratic field. Here we do the same for an arbitrary real quadratic field $K$ as base. As before, the action of $\mathrm{Gal}(K/\mathbb{Q})$ on the $p$-class tower group of $K$ plays a crucial role; however, the presence of units of infinite order in the ground field significantly complicates the possibilities for groups that can occur. We also sharpen our results in the imaginary quadratic field case by removing a certain hypothesis, using ideas of Boston and Wood. In an appendix, we show how the probabilities introduced for finite $p$-groups can be extended in a consistent way to the infinite pro-$p$ groups which can arise in both the real and imaginary quadratic settings.}

math.NT↗
arXiv2017

The $2$-Class Tower of $\mathbb{Q}(\sqrt{-5460})$

The seminal papers in the field of root-discriminant bounds are those of Odlyzko and Martinet. Both papers include the question of whether the field $\mathbb{Q}(\sqrt{-5460})$ has finite or infinite $2$-class tower. This is a critical case that will either substantially lower the best known upper bound for lim inf of root-discriminants (if infinite) or else give a counter-example to what is often termed Martinet's conjecture or question (if finite). Using extensive computation and introducing some new techniques, we give strong evidence that the tower is in fact finite, establishing other properties of its Galois group en route.

math.NT↗
arXiv2017

Exploiting Algebraic Structure in Global Optimization and the Belgian Chocolate Problem

The Belgian chocolate problem involves maximizing a parameter {\delta} over a non-convex region of polynomials. In this paper we detail a global optimization method for this problem that outperforms previous such methods by exploiting underlying algebraic structure. Previous work has focused on iterative methods that, due to the complicated non-convex feasible region, may require many iterations or result in non-optimal {\delta}. By contrast, our method locates the largest known value of {\delta} in a non-iterative manner. We do this by using the algebraic structure to go directly to large limiting values, reducing the problem to a simpler combinatorial optimization problem. While these limiting values are not necessarily feasible, we give an explicit algorithm for arbitrarily approximating them by feasible {\delta}. Using this approach, we find the largest known value of {\delta} to date, {\delta} = 0.9808348. We also demonstrate that in low degree settings, our method recovers previously known upper bounds on {\delta} and that prior methods converge towards the {\delta} we find.

math.OC↗
arXiv2017

The Weight Distribution of Quasi-quadratic Residue Codes

In this paper, we begin by reviewing some of the known properties of QQR codes and proved that $PSL_2(p)$ acts on the extended QQR code when $p \equiv 3 \pmod 4$. Using this discovery, we then showed their weight polynomials satisfy a strong divisibility condition, namely that they are divisible by $(x^2 + y^2)^{d-1}$, where $d$ is the corresponding minimum distance. Using this result, we were able to construct an efficient algorithm to compute weight polynomials for QQR codes and correct errors in existing results on quadratic residue codes. In the second half, we use the relation between the weight of codewords and the number of points on hyperelliptic curves to prove that the symmetrized distribution of a set of hyperelliptic curves is asymptotically normal.

cs.IT↗
arXiv2016

Nonabelian Cohen-Lenstra Heuristics over Function Fields

Boston, Bush, and Hajir have developed heuristics, extending the Cohen-Lenstra heuristics, that conjecture the distribution of the Galois groups of the maximal unramified pro-p extensions of imaginary quadratic number fields for p an odd prime. In this paper, we find the moments of their proposed distribution, and further prove there is a unique distribution with those moments. Further, we show that in the function field analog, for imaginary quadratic extensions of F_q(t), the Galois groups of the maximal unramified pro-p extensions, as q goes to infinity, have the moments predicted by the Boston, Bush, and Hajir heuristics. In fact, we determine the moments of the Galois groups of the maximal unramified pro-odd extensions of imaginary quadratic function fields, leading to a conjecture on Galois groups of the maximal unramified pro-odd extensions of imaginary quadratic number fields.

math.NT↗
arXiv2015

A Characterization of Deterministic Sampling Patterns for Low-Rank Matrix Completion

Low-rank matrix completion (LRMC) problems arise in a wide variety of applications. Previous theory mainly provides conditions for completion under missing-at-random samplings. This paper studies deterministic conditions for completion. An incomplete $d \times N$ matrix is finitely rank-$r$ completable if there are at most finitely many rank-$r$ matrices that agree with all its observed entries. Finite completability is the tipping point in LRMC, as a few additional samples of a finitely completable matrix guarantee its unique completability. The main contribution of this paper is a deterministic sampling condition for finite completability. We use this to also derive deterministic sampling conditions for unique completability that can be efficiently verified. We also show that under uniform random sampling schemes, these conditions are satisfied with high probability if $O(\max\{r,\log d\})$ entries per column are observed. These findings have several implications on LRMC regarding lower bounds, sample and computational complexity, the role of coherence, adaptive settings and the validation of any completion algorithm. We complement our theoretical results with experiments that support our findings and motivate future analysis of uncharted sampling regimes.

stat.ML↗
arXiv2014

On the Algebraic Structure of Linear Trellises

Trellises are crucial graphical representations of codes. While conventional trellises are well understood, the general theory of (tail-biting) trellises is still under development. Iterative decoding concretely motivates such theory. In this paper we first develop a new algebraic framework for a systematic analysis of linear trellises which enables us to address open foundational questions. In particular, we present a useful and powerful characterization of linear trellis isomorphy. We also obtain a new proof of the Factorization Theorem of Koetter/Vardy and point out unnoticed problems for the group case. Next, we apply our work to: describe all the elementary trellis factorizations of linear trellises and consequently to determine all the minimal linear trellises for a given code; prove that nonmergeable one-to-one linear trellises are strikingly determined by the edge-label sequences of certain closed paths; prove self-duality theorems for minimal linear trellises; analyze quasi-cyclic linear trellises and consequently extend results on reduced linear trellises to nonreduced ones. To achieve this, we also provide new insight into mergeability and path connectivity properties of linear trellises. Our classification results are important for iterative decoding as we show that minimal linear trellises can yield different pseudocodewords even if they have the same graph structure.

cs.IT↗
arXiv2013

A Refined Conjecture For Factoring Iterates Of Quadratic Polynomials Over Finite Fields

Jones and Boston conjectured that the factorization process for iterates of irreducible quadratic polynomials over finite fields is approximated by a Markov model. In this paper, we find unexpected and intricate behavior for some quadratic polynomials, in particular for the ones with tail size one. We also propose a multi-step Markov model that explains these new observations better than the model of Jones and Boston.

math.NT↗
arXiv2013

An infinite family of 2-groups with mixed Beauville structures

We construct an infinite family of triples $(G_k,H_k,T_k)$, where $G_k$ are 2-groups of increasing order, $H_k$ are index-2 subgroups of $G_k$, and $T_k$ are pairs of generators of $H_k$. We show that the triples $u_k = (G_k,H_k,T_k)$ are mixed Beauville structures if $k$ is not a power of 2. This is the first known infinite family of 2-groups admitting mixed Beauville structures. Moreover, the associated Beauville surface $S(u_3)$ is real and, for $k > 3$ not a power of 2, the Beauville surface $S(u_k)$ is not biholomorphic to $\bar{S(u_k)}$.

math.AG↗
arXiv2012

Random pro-p groups, braid groups, and random tame Galois groups

We introduce a heuristic prediction for the distribution of the isomorphism class of the Galois group of the maximal pro-p extension of Q unramified outside a "random" set of primes. This is guided by reasoning similar to that governing the Cohen-Lenstra conjectures. We conclude by describing theoretical and experimental evidence for our heuristic.

math.GR↗
arXiv2011

Heuristics for $p$-class towers of imaginary quadratic fields, with an Appendix by Jonathan Blackhurst

Cohen and Lenstra have given a heuristic which, for a fixed odd prime $p$, leads to many interesting predictions about the distribution of $p$-class groups of imaginary quadratic fields. We extend the Cohen-Lenstra heuristic to a non-abelian setting by considering, for each imaginary quadratic field $K$, the Galois group of the $p$-class tower of $K$, i.e. $G_K:=\mathrm{Gal}(K_\infty/K)$ where $K_\infty$ is the maximal unramified $p$-extension of $K$. By class field theory, the maximal abelian quotient of $G_K$ is isomorphic to the $p$-class group of $K$. For integers $c\geq 1$, we give a heuristic of Cohen-Lentra type for the maximal $p$-class $c$ quotient of $\G_K$ and thereby give a conjectural formula for how frequently a given $p$-group of $p$-class $c$ occurs in this manner. In particular, we predict that every finite Schur $\sigma$-group occurs as $G_K$ for infinitely many fields $K$. We present numerical data in support of these conjectures.

math.NT↗
arXiv2011

A Note on Beauville p-Groups

We examine which $p$-groups of order $\le p^6$ are Beauville. We completely classify them for groups of order $\le p^4$. We also show that the proportion of 2-generated groups of order $p^5$ which are Beauville tends to 1 as $p$ tends to infinity; this is not true, however, for groups of order $p^6$. For each prime $p$ we determine the smallest non-abelian Beauville $p$-group.

math.GR↗
arXiv2009

Pro-p groups and towers of rational homology spheres

In the preceding paper, Calegari and Dunfield exhibit a sequence of hyperbolic 3-manifolds which have increasing injectivity radius, and which, subject to some conjectures in number theory, are rational homology spheres. We prove unconditionally that these manifolds are rational homology spheres, and give a sufficient condition for a tower of hyperbolic 3-manifolds to have first Betti number 0 at each level. The methods involved are purely pro-p group theoretical.

math.GT↗