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Given smooth, projective, geometrically integral algebraic curves $X$ and $Y$ defined over a number field $K$, assuming that there is a non-constant $K$-morphism $\varphi \colon X \to Y$, we give an upper bound on the minimum of the degrees…
In the recent paper [AF12], we introduced an analysis of the Brylinski-Kostant model for spherical minimal representations for simple real Lie groups of non Hermitian type. We generalize here that analysis and give a unified geometric…
The determinantal variety $\Sigma_{pq}$ is defined to be the set of all $p\times q$ real matrices with $p\geq q$ whose ranks are strictly smaller than $q$. It is proved that $\Sigma_{pq}$ is a minimal cone in $\mathbb R^{pq}$ and all its…
In this paper we study the minimal faithful permutation representations of $SL_n(\mathbb F_q)$ and $GL_n(\mathbb F_q)$.
We generalize the following classical result of Fubini for pseudo-Riemannian metrics: if three essentially different metrics on $M^{n\ge 3}$ share the same unparametrized geodesics, and two of them (say, $g$ and $\bar g$) are strictly…
The $n^{th}$ cyclotomic polynomial $\Phi_n(x)$ is the minimal polynomial of an $n^{th}$ primitive root of unity. Its coefficients are the subject of intensive study and some formulas are known for them. Here we are interested in formulas…
We study the topological complexity of sets defined using Khovanskii's Pfaffian functions, in terms of an appropriate notion of format for those sets. We consider semi- and sub-Pfaffian sets, but more generally any definable set in the…
For an upper semi-continuous set-valued mapping from one topological space to another and for a lower semi-continuous function defined on the product of these spaces, Berge's theorem states lower semi-continuity of the minimum of this…
We investigate families of minimal rational curves on Schubert varieties, their Bott-Samelson desingularizations, and their generalizations constructed by Nicolas Perrin in the minuscule case. In particular, we describe the minimal families…
We classify binary minimal clones into seven categories: affine algebras, rectangular bands, $p$-cyclic groupoids, spirals, non-Taylor partial semilattices, melds, and dispersive algebras. Each category has nice enough properties to…
For a family $\mathcal{F}$ of sets and a disjoint pair $A,B$ we let $\mathcal{F}(A,\overline{B})=\{F\in \mathcal{F}: A\subseteq F, ~B\cap F=\emptyset\}$. The \textbf{$(p,q)$-d\"omd\"od\"om} of a family $\mathcal{F}\subseteq 2^{[n]}$ is…
The aim of this paper is to provide an explicit basis of the miniversal deformation of a monomial curve defined by a free semigroup -- these curves make up a notable family of complete intersection monomial curves. First, we dispense a…
We characterize the curves in $ P^3$ which are minimal in their biliaison class. Such curves are exactly the curves which do no admit an elementary descending biliaison. As a consequence we have that every curve in $P^3$ can be obtained…
We give an explicit description of the minimal regular model of bihyperelliptic curves with semistable reduction over a local field of odd residue characteristic. We do this using a generalisation of the cluster picture; a completely…
Let $q$ be a prime power. We determine all permutation trinomials of $\Bbb F_{q^2}$ of the form $ax+bx^q+x^{2q-1}\in\Bbb F_{q^2}[x]$. The subclass of such permutation trinomials of $\Bbb F_{q^2}$ with $a,b\in\Bbb F_q$ was determined in a…
A binary modified de Bruijn sequence is an infinite and periodic binary sequence derived by removing a zero from the longest run of zeros in a binary de Bruijn sequence. The minimal polynomial of the modified sequence is its unique…
A partial order on the set of prime knots can be defined by the existence of an epimorphism between knot groups. We prove that all the prime knots with up to $6$ crossings are minimal. We also show that each fibered knot with the…
Given a subset $\mathcal{S}\subseteq \mathbb{F}_q[x]$ and fixed integers $n,m\in \mathbb{N}$, we study the distribution of the smallest denominator $Q\in \mathcal{S}$ for which there exists $\mathbf{P}\in \mathbb{F}_q[x]^m$ such that…
A subset $M \subset \textbf{R}^3$ is called a \emph{basic subset}, if for any funciton $f \colon M \to \textbf{R}$ there exist such functions $f_1; f_2; f_3 \colon \textbf{R} \to \textbf{R}$ that $f(x_1, x_2, x_3) = f_1(x_1) + f_2(x_2) +…
For $n\ge 3$ let $f(n)$ be the least positive integer $k$ such that $\binom nk>\frac{2^n}{n+1}$. In this paper we investigate the properties of $f(n)$.