arXiv subjects
Publications and source records attributed to Jamie Pommersheim.
We give a simple and complete description of those convex lattice polygons in the plane that can be dissected into lattice triangles of integer area. A new version of Sperner's Lemma plays a central role.
Given an $n\times n$ matrix with integer entries in the range $[-h,h]$, how close can two of its distinct eigenvalues be? The best previously known examples have a minimum gap of $h^{-O(n)}$. Here we give an explicit construction of matrices with entries in $[0,h]$ with two eigenvalues separated by at most $h^{-n^2/16+o(n^2)}$. Up to a constant in the exponent, this agrees with the known lower bound of $Ω((2\sqrt{n})^{-n^2}h^{-n^2})$ \cite{mahler1964inequality}. Bounds on the minimum gap are relevant to the worst case analysis of algorithms for diagonalization and computing canonical forms of integer matrices. In addition to our explicit construction, we show there are many matrices with a slightly larger gap of roughly $h^{-n^2/32}$. We also construct 0-1 matrices which have two eigenvalues separated by at most $2^{-n^2/64+o(n^2)}$.
Given a trapezoid dissected into triangles, the area of any triangle determined by either diagonal of the trapezoid is integral over the ring generated by the areas of the triangles in the dissection. Given a parallelogram dissected into triangles, the area of any one of the triangles of the dissection is integral over the ring generated by the areas of the other triangles. In both cases, the integrality relations are invariant under deformation of the dissection. The trapezoid theorem implies and provides a new context for Monsky's Equidissection Theorem that a square cannot be dissected into an odd number of triangles of equal area. A corollary of these results is that the area polynomials for parallelograms introduced in previous work have all leading coefficients equal to $\pm 1$.
We study the query complexity of quantum learning problems in which the oracles form a group $G$ of unitary matrices. In the simplest case, one wishes to identify the oracle, and we find a description of the optimal success probability of a $t$-query quantum algorithm in terms of group characters. As an application, we show that $Ω(n)$ queries are required to identify a random permutation in $S_n$. More generally, suppose $H$ is a fixed subgroup of the group $G$ of oracles, and given access to an oracle sampled uniformly from $G$, we want to learn which coset of $H$ the oracle belongs to. We call this problem coset identification and it generalizes a number of well-known quantum algorithms including the Bernstein-Vazirani problem, the van Dam problem and finite field polynomial interpolation. We provide character-theoretic formulas for the optimal success probability achieved by a $t$-query algorithm for this problem. One application involves the Heisenberg group and provides a family of problems depending on $n$ which require $n+1$ queries classically and only $1$ query quantumly.
Every set of natural numbers determines a generating function convergent for $q \in (-1,1)$ whose behavior as $q \rightarrow 1^-$ determines a germ. These germs admit a natural partial ordering that can be used to compare sets of natural numbers in a manner that generalizes both cardinality of finite sets and density of infinite sets. For any finite set $D$ of positive integers, call a set $S$ "$D$-avoiding" if no two elements of $S$ differ by an element of $D$. We study the problem of determining, for fixed $D$, all $D$-avoiding sets that are maximal in the germ order. In many cases, we can show that there is exactly one such set. We apply this to the study of one-dimensional packing problems.
Monsky's celebrated equidissection theorem follows from his more general proof of the existence of a polynomial relation $f$ among the areas of the triangles in a dissection of the unit square. More recently, the authors studied a different polynomial $p$, also a relation among the areas of the triangles in such a dissection, that is invariant under certain deformations of the dissection. In this paper we study the relationship between these two polynomials. We first generalize the notion of dissection, allowing triangles whose orientation differs from that of the plane. We define a deformation space of these generalized dissections and we show that this space is an irreducible algebraic variety. We then extend the theorem of Monsky to the context of generalized dissections, showing that Monsky's polynomial $f$ can be chosen to be invariant under deformation. Although $f$ is not uniquely defined, the interplay between $p$ and $f$ then allows us to identify a canonical pair of choices for the polynomial $f$. In many cases, all of the coefficients of the canonical $f$ polynomials are positive. We also use the deformation-invariance of $f$ to prove that the polynomial $p$ is congruent modulo 2 to a power of the sum of its variables.