Fattening up Warning's Second Theorem

We present a generalization of Warning's Second Theorem to polynomial systems over a finite local principal ring with suitably restricted input and output variables. This generalizes a recent result with Forrow and Schmitt (and gives a new proof of that result). Applications to additive group theory, graph theory and polynomial interpolation are pursued in detail.Comment: 21 page

Bases and Structure Constants of Generalized Splines with Integer Coefficients on Cycles

An integer generalized spline is a set of vertex labels on an edge-labeled graph that satisfy the condition that if two vertices are joined by an edge, the vertex labels are congruent modulo the edge label. Foundational work on these objects comes from Gilbert, Polster, and Tymoczko, who generalize ideas from geometry/topology (equivariant cohomology rings) and algebra (algebraic splines) to develop the notion of generalized splines. Gilbert, Polster, and Tymoczko prove that the ring of splines on a graph can be decomposed in terms of splines on its subgraphs (in particular, on trees and cycles), and then fully analyze splines on trees. Following Handschy-Melnick-Reinders and Rose, we analyze splines on cycles, in our case integer generalized splines. The primary goal of this paper is to establish two new bases for the module of integer generalized splines on cycles: the triangulation basis and the King basis. Unlike bases in previous work, we are able to characterize each basis element completely in terms of the edge labels of the underlying cycle. As an application we explicitly construct the multiplication table for the ring of integer generalized splines in terms of the King basis.Comment: 18 page

Lattice homomorphisms between weak orders

We classify surjective lattice homomorphisms $W\to W'$ between the weak orders on finite Coxeter groups. Equivalently, we classify lattice congruences $\Theta$ on $W$ such that the quotient $W/\Theta$ is isomorphic to $W'$. Surprisingly, surjective homomorphisms exist quite generally: They exist if and only if the diagram of $W'$ is obtained from the diagram of $W$ by deleting vertices, deleting edges, and/or decreasing edge labels. A surjective homomorphism $W\to W'$ is determined by its restrictions to rank-two standard parabolic subgroups of $W$. Despite seeming natural in the setting of Coxeter groups, this determination in rank two is nontrivial. Indeed, from the combinatorial lattice theory point of view, all of these classification results should appear unlikely a priori. As an application of the classification of surjective homomorphisms between weak orders, we also obtain a classification of surjective homomorphisms between Cambrian lattices and a general construction of refinement relations between Cambrian fans.Comment: 45 pages, about 5 of which are taken up by figures. Version 2: Minor expository changes, in the abstract and introduction only. Version 3: Added uniform Lie-theoretic proof of root system containment result for Kac-Moody root systems. Version 4: Added citation for the proof added in version 3. Version 5: Final version (modulo formatting) to appear in the Electronic Journal of Combinatoric

Congruence lattices of ideals in categories and (partial) semigroups

This paper presents a unified framework for determining the congruences on a number of monoids and categories of transformations, diagrams, matrices and braids, and on all their ideals. The key theoretical advances present an iterative process of stacking certain normal subgroup lattices on top of each other to successively build congruence lattices of a chain of ideals. This is applied to several specific categories of: transformations; order/orientation preserving/reversing transformations; partitions; planar/annular partitions; Brauer, Temperley--Lieb and Jones partitions; linear and projective linear transformations; and partial braids. Special considerations are needed for certain small ideals, and technically more intricate theoretical underpinnings for the linear and partial braid categories.Comment: V2 incorporates referee's suggestions. To appear in Memoirs of the AMS. 108 pages, 32 figure

Covering systems with restricted divisibility

We prove that every distinct covering system has a modulus divisible by either 2 or 3.Comment: Light change

Analogues of the Jordan-Holder theorem for transitive G-sets

Let G be a transitive group of permutations of a finite set X, and suppose that some element of G has at most two orbits on X. We prove that any two maximal chains of groups between G and a point-stabilizer of G have the same length, and the same sequence of relative indices between consecutive groups (up to permutation). We also deduce the same conclusion when G has a transitive quasi-Hamiltonian subgroup

An exploration of Nathanson's gg-adic representations of integers

We use Nathanson's $g$-adic representation of integers to relate metric properties of Cayley graphs of the integers with respect to various infinite generating sets $S$ to problems in additive number theory. If $S$ consists of all powers of a fixed integer $g$, we find explicit formulas for the smallest positive integer of a given length. This is related to finding the smallest positive integer expressible as a fixed number of sums and differences of powers of $g$. We also consider $S$ to be the set of all powers of all primes and bound the diameter of Cayley graph by relating it to Goldbach's conjecture.Comment: 10 pages, rewritten to replace "On locally infinite Cayley graphs of Z.

Mesoprimary decomposition of binomial submodules

Recent results of Kahle and Miller give a method of constructing primary decompositions of binomial ideals by first constructing "mesoprimary decompositions" determined by their underlying monoid congruences. These mesoprimary decompositions are highly combinatorial in nature, and are designed to parallel standard primary decomposition over Noetherean rings. In this paper, we generalize mesoprimary decomposition from binomial ideals to "binomial submodules" of certain graded modules over the corresponding monoid algebra, analogous to the way primary decomposition of ideals over a Noetherean ring $R$ generalizes to $R$-modules. The result is a combinatorial method of constructing primary decompositions that, when restricting to the special case of binomial ideals, coincides with the method introduced by Kahle and Miller

Explicit moduli spaces for congruences of elliptic curves

We determine explicit birational models over Q for the modular surfaces parametrising pairs of N-congruent elliptic curves in all cases where this surface is an elliptic surface. In each case we also determine the rank of the Mordell-Weil lattice and the geometric Picard number.Comment: 20 page

A generalization of Sch\"{o}nemann's theorem via a graph theoretic method

Recently, Grynkiewicz et al. [{\it Israel J. Math.} {\bf 193} (2013), 359--398], using tools from additive combinatorics and group theory, proved necessary and sufficient conditions under which the linear congruence $a_1x_1+\cdots +a_kx_k\equiv b \pmod{n}$, where $a_1,\ldots,a_k,b,n$ ($n\geq 1$) are arbitrary integers, has a solution $\langle x_1,\ldots,x_k \rangle \in \Z_{n}^k$ with all $x_i$ distinct. So, it would be an interesting problem to give an explicit formula for the number of such solutions. Quite surprisingly, this problem was first considered, in a special case, by Sch\"{o}nemann almost two centuries ago(!) but his result seems to have been forgotten. Sch\"{o}nemann [{\it J. Reine Angew. Math.} {\bf 1839} (1839), 231--243] proved an explicit formula for the number of such solutions when $b=0$, $n=p$ a prime, and $\sum_{i=1}^k a_i \equiv 0 \pmod{p}$ but $\sum_{i \in I} a_i \not\equiv 0 \pmod{p}$ for all $\emptyset \not= I\varsubsetneq \lbrace 1, \ldots, k\rbrace$. In this paper, we generalize Sch\"{o}nemann's theorem using a result on the number of solutions of linear congruences due to D. N. Lehmer and also a result on graph enumeration. This seems to be a rather uncommon method in the area; besides, our proof technique or its modifications may be useful for dealing with other cases of this problem (or even the general case) or other relevant problems