Searcharxiv⌕ Search

arXiv subjects

Henry Towsner

Publications and source records attributed to Henry Towsner.

At least 37 records · Page 2Linked to original sources
arXiv2019

Explicit polynomial bounds on prime ideals in polynomial rings over fields

Suppose $I$ is an ideal of a polynomial ring over a field, $I\subseteq k[x_1,\ldots,x_n]$, and whenever $fg\in I$ with degree $\leq b$, then either $f\in I$ or $g\in I$. When $b$ is sufficiently large, it follows that $I$ is prime. Schmidt-Göttsch proved that "sufficiently large" can be taken to be a polynomial in the degree of generators of $I$ (with the degree of this polynomial depending on $n$). However Schmidt-Göttsch used model-theoretic methods to show this, and did not give any indication of how large the degree of this polynomial is. In this paper we obtain an explicit bound on $b$, polynomial in the degree of the generators of $I$. We also give a similar bound for detecting maximal ideals in $k[x_1,\ldots,x_n]$.

math.AC↗
arXiv2018

Proof mining and effective bounds in differential polynomial rings

Using the functional interpretation from proof theory, we analyze nonconstructive proofs of several central theorems about polynomial and differential polynomial rings. We extract effective bounds, some of which are new to the literature, from the resulting proofs. In the process we discuss the constructive content of Noetherian rings and the Nullstellensatz in both the classical and differential settings. Sufficient background is given to understand the proof-theoretic and differential-algebraic framework of the main results.

math.LO↗
arXiv2018

Constructing Sequences One Step at a Time

We propose a new method for constructing Turing ideals satisfying principles of reverse mathematics below the Chain-Antichain Principle (CAC). Using this method, we are able to prove several new separations in the presence of Weak Konig's Lemma (WKL), including showing that CAC+WKL does not imply the thin set theorem for pairs, and that the principle "the product of well-quasi-orders is a well-quasi-order" is strictly between CAC and the Ascending/Descending Sequences principle, even in the presence of WKL.

math.LO↗
arXiv2018

A short nonalgorithmic proof of the containers theorem for hypergraphs

Recently the breakthrough method of hypergraph containers, developed independently by Balogh, Morris, and Samotij as well as Saxton and Thomason, has been used to study sparse random analogs of a variety of classical problems from combinatorics and number theory. The previously known proofs of the containers theorem use the so-called scythe algorithm---an iterative procedure that runs through the vertices of the hypergraph. (Saxton and Thomason have also proposed an alternative, randomized construction in the case of simple hypergraphs.) Here we present the first known deterministic proof of the containers theorem that is not algorithmic, i.e., it does not involve an iterative process. Our proof is less than 4 pages long while being entirely self-contained and conceptually transparent. Although our proof is completely elementary, it was inspired by considering hypergraphs in the setting of nonstandard analysis, where there is a notion of dimension capturing the logarithmic rate of growth of finite sets. Before presenting the proof in full detail, we include a one-page informal outline that refers to this notion of dimension and summarizes the essence of the argument.

math.CO↗
arXiv2018

Erdos-Moser and ISigma_2

The first-order part of the Ramsey's Theorem for pairs with an arbitrary number of colors is known to be precisely BSigma03. We compare this to the known division of Ramsey's Theorem for pairs into the weaker principles, EM (the Erdős-Moser principle) and ADS (the ascending-descending sequence principle): we show that the additional strength beyond ISigma02 is entirely due to the arbitrary color analog of ADS. Specifically, we show that ADS for an arbitrary number of colors implies BSigma03 while EM for an arbitrary number of colors is Pi11-conservative over ISigma02 and it does not imply ISigma02.

math.LO↗
arXiv2018

An Analytic Approach to Sparse Hypergraphs: Hypergraph Removal

The use of tools from analysis to approach problems in graph theory has become an active area of research. Usually such methods are applied to problems involving dense graphs and hypergraphs; here we give the an extension of such methods to sparse but pseudorandom hypergraphs. We use this framework to give a proof of hypergraph removal for sub-hypergraphs of sparse random hypergraphs.

math.CO↗
arXiv2017

Nonstandard Convergence Gives Bounds on Jumps

If we know that some kind of sequence always converges, we can ask how quickly and how uniformly it converges. Many convergent sequences converge non-uniformly and, relatedly, have no computable rate of convergence. However proof-theoretic ideas often guarantee the existence of a uniform "meta-stable" rate of convergence. We show that obtaining a stronger bound---a uniform bound on the number of jumps the sequence makes---is equivalent to being able to strengthen convergence to occur in the nonstandard numbers. We use this to obtain bounds on the number of jumps in nonconventional ergodic averages.

math.LO↗
arXiv2017

More or Less Uniform Convergence

Uniform metastable convergence is a weak form of uniform convergence for a family of sequences. In this paper we explore the way that metastable convergence stratifies into a family of notions indexed by countable ordinals. We give two versions of this stratified family, loosely speaking, they correspond to the model theoretic and proof theoretic perspectives. For the model theoretic version, which we call abstract alpha-uniform convergence, we show that uniform metastable convergence is equivalent to abstract $α$-uniform convergence for some alpha, and that abstract omega-uniform convergence is equivalent to uniformly bounded oscillation of the family of sequences. The proof theoretic version, which we call concrete alpha-uniform convergence, is less canonical (it depends on a choice of ordinal notation), but appears naturally when "proof mining" convergence proofs to obtain quantitative bounds. We show that these hierarchies are strict by exhibiting a family of which is concretely alpha+1-uniformly convergent but not abstractly alpha-uniformly convergent for each alpha<omega_1.

math.LO↗
arXiv2017

Explicit sentences distinguishing McDuff's II$_1$ factors

Recently, Boutonnet, Chifan, and Ioana proved that McDuff's examples of continuum many pairwise non-isomorphic separable II$_1$ factors are in fact pairwise non-elementarily equivalent. Their proof proceeded by showing that any ultrapowers of any two distinct McDuff examples are not isomorphic. In a paper by the first two authors of this paper, Ehrenfeucht-Fraïsse games were used to find an upper bound on the quantifier complexity of sentences distinguishing the McDuff examples, leaving it as an open question to find concrete sentences distinguishing the McDuff factors. In this paper, we answer this question by providing such concrete sentences.

math.LO↗
arXiv2016

The structure of combinatorial Markov processes

Every exchangeable Feller process taking values in a suitably nice combinatorial state space can be constructed by a system of iterated random Lipschitz functions. In discrete time, the construction proceeds by iterated application of independent, identically distributed functions, while in continuous time the random functions occur as the atoms of a time homogeneous Poisson point process. We further show that every exchangeable Feller process projects to a Feller process in an appropriate limit space, akin to the projection of partition-valued processes into the ranked-simplex and graph-valued processes into the space of graph limits. Together, our main theorems establish common structural features shared by all exchangeable combinatorial Feller processes, regardless of the dynamics or resident state space, thereby generalizing behaviors previously observed for exchangeable coalescent and fragmentation processes as well as other combinatorial stochastic processes. If, in addition, an exchangeable Feller process evolves on a state space satisfying the $n$-disjoint amalgamation property for all $n\geq1$, then its jump measure can be decomposed explicitly in the sense of Lévy--Itô--Khintchine.

math.PR↗
arXiv2016

Relative exchangeability with equivalence relations

We describe an Aldous--Hoover-type characterization of random relational structures that are exchangeable relative to a fixed structure which may have various equivalence relations. Our main theorem gives the common generalization of the results on relative exchangeability due to Ackerman \cite{Ackerman2015} and Crane and Towsner \cite{CraneTowsner2015} and hierarchical exchangeability results due to Austin and Panchenko \cite{AustinPanchenko2014}.

math.LO↗
arXiv2015

Relatively exchangeable structures

We study random relational structures that are \emph{relatively exchangeable}---that is, whose distributions are invariant under the automorphisms of a reference structure $\mathfrak{M}$. When $\mathfrak{M}$ has {\em trivial definable closure}, every relatively exchangeable structure satisfies a general Aldous--Hoover-type representation. If $\mathfrak{M}$ satisfies the stronger properties of {\em ultrahomogeneity} and {\em $n$-disjoint amalgamation property} ($n$-DAP) for every $n\geq1$, then relatively exchangeable structures have a more precise description whereby each component depends locally on $\mathfrak{M}$.

math.LO↗
arXiv2015

Epsilon Substitution for $ID_1$ via Cut-Elimination

The $ε$-substitution method is a technique for giving consistency proofs for theories of arithmetic. We use this technique to give a proof of the consistency of the impredicative theory $ID_1$ using a variant of the cut-elimination formalism introduced by Mints.

math.LO↗
arXiv2015

Computable Ramsey's Theorem for Pairs Needs Infinitely Many Pi-0-2 Sets

In \cite{J}, Theorem 4.2, Jockusch proves that for any computable k-coloring of pairs of integers, there is an infinite $Π^0_2$ homogeneous set. The proof uses a countable collection of $Π^0_2$ sets as potential infinite homogeneous sets. In a remark preceding the proof, Jockusch states without proof that it can be shown that there is no computable way to prove this result with a finite number of $Π^0_2$ sets. We provide a proof of this latter fact.

math.LO↗
arXiv2015

A Worked Example of the Functional Interpretation

The functional interpretation is a systematic, syntactic method for transforming certain non-constructive proofs into constructive proofs with explicit bounds. We illustrate the interpretation by working through a concrete, fairly simple example, with almost no reference to formal logic, and then explain the connection with the underlying proof-theoretic methods.

math.LO↗
arXiv2015

Limits of Sequences of Markov Chains

We study the limiting object of a sequence of Markov chains analogous to the limits of graphs, hypergraphs, and other objects which have been studied. Following a suggestion of Aldous, we assign to a sequence of finite Markov chains with bounded mixing times a unique limit object: an infinite Markov chain with a measurable state space. The limits of the Markov chains we consider have discrete spectra, which makes the limit theory simpler than the general graph case, and illustrates how the discrete spectrum setting (sometimes called "random-free" or "product measurable") is simpler than the general case.

math.LO↗