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Squares in Polynomial Product Sequences
Squarefull numbers Polynomial Product Sequences Number Theory
2011/8/31
Abstract: Let F(n) be a polynomial of degree at least 2 with integer coefficients. We consider the products N_x=\prod_{1 \le n \le x} F(n) and show that N_x should only rarely be a perfect power. In p...
A polynomial bound on solutions of quadratic equations in free groups
polynomial bound quadratic equations free groups Group Theory
2011/8/31
Abstract: We provide polynomial upper bounds on the size of a shortest solution for quadratic equations in a free group. A similar bound is given for parametric solutions in the description of solutio...
The Kauffman polynomial and trivalent graphs
braids invariants for graphs and links Kau man polynomial knotted graphs
2011/8/26
Abstract: We construct a state model for the two-variable Kauffman polynomial using planar trivalent graphs. We also use this model to obtain a polynomial invariant for trivalent graphs embedded in th...
The classification of polynomial basins of infinity
polynomial basins of infinity Dynamical Systems Complex Variables
2011/8/26
Abstract: We consider the problem of classifying the dynamics of complex polynomials $f: \mathbb{C} \to \mathbb{C}$ restricted to their basins of infinity. We synthesize existing combinatorial tools -...
Polynomial families of tautological classes on $\mathcal{M}_{g,n}^{rt}$
Polynomial families tautological classes Algebraic Geometry
2011/8/25
Abstract: We study classes $P_{g,T}(\alpha;\beta)$ on the moduli space of stable, genus g curves with rational tails defined by pushing forward the virtual fundamental classes of spaces of relative st...
Ehrhart's polynomial for equilateral triangles in $\mathbb Z^3$
Ehrhart polynomial linear Diophantine equations lattice points equilateral triangles sequences regular integral polytopes
2011/8/25
Abstract: In this paper we calculate the Ehrhart's polynomial associated with a 2-dimensional regular polytope (i.e. equilateral triangles) in $\mathbb Z^3$. The polynomial takes a relatively simple f...
Abstract: The degree chromatic polynomial $Pm(G,k)$ of a graph $G$ counts the number of $k$-colorings in which no vertex has $m$ adjacent vertices of its same color. We prove Humpert and Martin's conj...
Quasi-polynomial mixing of the 2D stochastic Ising model with "plus" boundary up to criticality
Ising model Mixing time Phase coexistence Glauber dynamics
2011/1/18
We considerably improve upon the recent result of [34] on the mixing time of Glauber dynamics for the 2d Ising model in a box of side L at low temperature and with random boundary conditions whose dis...
Polytopic Invariant Verification and Synthesis for Polynomial Dynamical Systems via Linear Programming
Polytopic Invariant Verification Synthesis for Polynomial Dynamical Systems Linear Programming
2011/1/18
This paper deals with the veri cation and the synthesis of polytopic invariant sets for
polynomial dynamical systems. An invariant set of a dynamical system is a subset of the state
space such that ...
A remark on Alexander polynomial criterion for bi-orderability of fibered 3-manifold groups
Bi-orderable group twisted Alexander polynomial
2011/1/17
We observe that Clay-Rolfsen’s obstruction of bi-orderability, which uses the classical Alexander polynomial, is not strengthened by using the twisted Alexander polynomials for finite representations ...
Tropical varieties with polynomial weights and corner loci of piecewise polynomials
Tropical varieties polynomial weights corner loci of piecewise polynomials
2011/2/28
Counting Euler characteristics of the discriminant of the quadratic equation in terms of
Newton polytopes in two different ways, G. Gusev ([Gus]) found an unexpected relation for
mixed volumes of tw...
A generalization of The Dress construction for a Tambara functor, and polynomial Tambara functors
A generalization Dress construction Tambara functor polynomial Tambara functors
2011/1/19
For a finite group G, (semi-)Mackey functors and (semi-)Tambara functors are regarded as G-bivariant analogs of (semi-)groups and (semi-)rings respectively. In fact if G is trivial, they agree with th...
On Stably free modules over Laurent polynomial rings
Stably free modules Hermite rings Unimodular rows
2011/2/21
We prove constructively that for any finite-dimensional commu-tative ring R and n dim(R) + 2, the group En(R[X,X−1]) acts transitively on Umn(R[X,X−1]). In particular, we obtain that for...
The cycolotomic polynomial topologically
Cyclotomic polynomial higher-dimensional tree matroid duality
2011/1/19
We interpret the coefficients of the cyclotomic polynomial in terms of simplicial homology.
The scope of refinable functions in wavelet theory is focused to localized functions. In our
paper we like to widen that scope, in particular we show that all polynomial functions are refinable. This...