Related papers: On the Pierce-Birkhoff Conjecture for Smooth Affin…

One of the major open problems in complexity theory is proving super-logarithmic lower bounds on the depth of circuits (i.e., $\mathbf{P}\not\subseteq\mathbf{NC}^1$). Karchmer, Raz, and Wigderson (Computational Complexity 5(3/4), 1995)…

A key property of an algebraic variety is whether it is absolutely irreducible, meaning that it remains irreducible over the algebraic closure of its defining field, and determining absolute irreducibility is important in algebraic geometry…

We prove several surjectivity criteria for $p$-adic representations. In particular, we classify all adjoint and simply connected group schemes $G$ over the Witt ring $W(k)$ of a finite field $k$ such that the epimorphism…

A geometric argument is given to prove that the Seifert genus of a positive knot equals its slice genus. A combinatorial invariant, giving a lower bound for the slice genus, is formulated for arbitrary knots. Properties and applications of…

We consider the finite W-superalgebras for a basic classical Lie superalgebra g associated with an even nilpotent element in g both over the field of complex numbers field and and over a filed of positive characteristic. We present the PBW…

The paper extends Birkhoff's theorem on doubly stochastic matrices to some countable families of discrete probability spaces with nonempty intersections. We join every two elements lying in the same probability space by an edge and…

The inverse conjecture for the Gowers norms $U^d(V)$ for finite-dimensional vector spaces $V$ over a finite field $\F$ asserts, roughly speaking, that a bounded function $f$ has large Gowers norm $\|f\|_{U^d(V)}$ if and only if it…

Invariance with respect to linear or affine transformations of the domain is arguably the most common symmetry exhibited by natural algebraic properties. In this work, we show that any low complexity affine-invariant property of…

We consider Reeb flows on the tight $3$-sphere admitting a pair of closed orbits forming a Hopf link. If the rotation numbers associated to the transverse linearized dynamics at these orbits fail to satisfy a certain resonance condition…

Under suitable hypotheses, we prove that a form of a projective homogeneous variety $G/P$ defined over the function field of a surface over an algebraically closed field has a rational point. The method uses an algebro-geometric analogue of…

For a branched cover between two closed orientable surfaces, the Riemann-Hurwitz formula relates the Euler characteristics of the surfaces, the total degree of the cover, and the total length of the partitions of the degree given by the…

Let $q$ be a power of a prime, let $\mathbb{F}_q$ be the finite field with $q$ elements and let $n \geq 2$. For a polynomial $h(x) \in \mathbb{F}_q[x]$ of degree $n \in \mathbb{N}$ and a subset $W \subseteq [0,n] := \{0, 1, \ldots, n\}$, we…

Garret Birkhoff's HSP theorem characterizes the classes of models of algebraic theories as those being closed with respect to homomorphic images, subalgebras, and products. In particular, it implies that an algebra $\mathbf{B}$ satisfies…

Sommese has conjectured a classification of smooth projective varieties X containing, as an ample divisor, a P^d-bundle Y over a smooth variety Z. This conjecture is known if d>1, if dim(X)<5, or if Z admits a finite morphism to an Abelian…

In this short note, we will show the following weak evidence of S. Lang conjecture over function fields. Let f : X ---> Y be a projective and surjective morphism of algebraic varieties over an algebraically closed field k of characteristic…

We develop an elementary method for proving the PBW theorem for associative algebras with an ascending filtration. The idea is roughly the following. At first, we deduce a proof of the PBW property for the {\it ascending} filtration (with…

We prove a generalization of the Poincar\'e-Birkhoff theorem for the open annulus showing that if a homeomorphism satisfies a certain twist condition and the nonwandering set is connected, then there is a fixed point. Our main focus is the…

We prove some new instances of a conjecture of Bachoc, Couvreur and Z\'emor that generalizes Freiman's $3k-4$ Theorem to a multiplicative version in a function field setting. As a consequence we find that if $F$ is a rational function field…

We study algebraic dynamical systems (and, more generally, $\sigma$-varieties) $\Phi:{\mathbb A}^n_{\mathbb C} \to {\mathbb A}^n_{\mathbb C}$ given by coordinatewise univariate polynomials by refining a theorem of Ritt. More precisely, we…

We extend the classical Poincar\'e-Birkhoff-Witt theorem to higher algebra by establishing a version that applies to spectral Lie algebras. We deduce this statement from a basic relation between operads in spectra: the commutative operad is…