Related papers: On homomorphisms from the Hamming cube to {\bf Z}

Let F_{n+m} be the free group of rank n+m, with generators x_1,...,x_{n+m}. An automorphism \phi of F_{n+m} is called partially symmetric if for each 1 \le i \le m, \phi(x_i) is conjugate to x_j or x_j^{-1} for some 1 \le j \le m. Let…

The hammock localization provides a model for a homotopy function complex in any Quillen model category. We prove that a homotopy between a pair of morphisms induces a homotopy between the maps induced by taking the hammock localization. We…

Let R be a commutative ring with 1. We prove that every homogeneous polynomial f(x_0,x_1,x_2) in R[x_0,x_1,x_2] up to degree 5 admits a linear Pfaffian R-representation. We believe that conceptually we give the shortest self-contained proof…

Let $\Omega$ be a compact subset of $\mathbb{C}$ and let $A$ be a unital simple, separable $C^*$-algebra with stable rank one, real rank zero and strict comparison. We show that, given a Cu-morphism $\alpha:{\rm Cu}(C(\Omega))\to {\rm…

Let $F(x)=(f_1(x), \dots, f_m(x))$ be such that $1, f_1, \dots, f_m$ are linearly independent polynomials with real coefficients. Based on ideas of Bachoc, DeCorte, Oliveira and Vallentin in combination with estimating certain oscillatory…

It is known due to Baker and Montgomery that almost all Fekete polynomials under certain ordering have at least one zero on the interval (0, 1). In terms of the positive-definiteness, Fekete polynomial has no zero on the interval (0, 1) if…

We prove a suite of results classifying holomorphic maps between configuration spaces of Riemann surfaces; we consider both the ordered and unordered setting as well as the cases of genus zero, one, and at least two. We give a complete…

Let $f$ and $g$ be Boolean functions over a finite Abelian group $\mathcal{G}$, where $g$ is fully known, and we have {\em query access} to $f$, that is, given any $x \in \mathcal{G}$ we can get the value $f(x)$. We study the tolerant…

We consider the problem of uniform sampling of points on an algebraic variety. Specifically, we develop a randomized algorithm that, given a small set of multivariate polynomials over a sufficiently large finite field, produces a common…

We show that the universal measuring coalgebras between Frobenius algebras turn the category of Frobenius algebras into a Hopf category (in the sense of Batista-Caenepeel-Vercruysse), and the universal comeasuring algebras between Frobenius…

In this paper, we prove a general principle of lifting an automorphism from positive characteristic to zero characteristic. We based on the principle to prove the automorphism group of Fano variety of cubic threefold (fourfold) acts on its…

Let $f$ be a holomorphic curve in $\mathbb{P}^n({\mathbb{C}})$ and let $\mathcal{D}=\{D_1,\ldots,D_q\}$ be a family of moving hypersurfaces defined by a set of homogeneous polynomials $\mathcal{Q}=\{Q_1,\ldots,Q_q\}$. For $j=1,\ldots,q$,…

A result of Lehrer describes a beautiful relationship between topological and combinatorial data on certain families of varieties with actions of finite reflection groups. His formula relates the cohomology of complex varieties to point…

Generalizing a recent result of Mann, we show that there is an explicit function $f:\left(0,1\right]\rightarrow\left(0,1\right]$ such that for every reduced word $w$, say in $d$ variables, there is an explicit reduced word $v$ in at most…

A group, $\fl{H}$, of automorphisms of a totally disconnected locally compact group, $G$, is flat if there is a compact open $U\leq G$ such that the index $[\alpha(U):U\cap \alpha(U)]$ is mininimized for every $\alpha\in\fl{H}$. The…

We prove an analogue of the classical Bateman-Horn conjecture on prime values of polynomials for the ring of polynomials over a large finite field. Namely, given non-associate, irreducible, separable and monic (in the variable $x$)…

We study the probability that a random polynomial with integer coefficients is reducible when factored over the rational numbers. Using computer-generated data, we investigate a number of different models, including both monic and non-monic…

The homotopy theory of representations of nets of algebras over a (small) category with values in a closed symmetric monoidal model category is developed. We illustrate how each morphism of nets of algebras determines a change-of-net…

Many natural combinatorial quantities can be expressed by counting the number of homomorphisms to a fixed relational structure. For example, the number of 3-colorings of an undirected graph $G$ is equal to the number of homomorphisms from…

We show that algebraic analogues of universal group covers, surjective group homomorphisms from a $\mathbb{Q}$-vector space to $F^{\times}$ with "standard kernel", are determined up to isomorphism of the algebraic structure by the…