Let $G$ be a group corresponding to a finite Coxeter system $(W,S)$. Does there exist an algorithm, which on input $(W,S)$ tells what is the minimal cardinality of a generating set for $G$? A weaker form of the question: does there exist an algorithm to check whether $G$ can be generated by 2 e...

Let $a$ be a non-zero integer. Consider the elliptic curve $E_a/\mathbb{Q}$ given by the equation $$ E_a: y^2 = x^3 + a. $$ For a cube-free integer $D$, define the elliptic curves $E_{aD^2}/\mathbb{Q}$ and $E_{aD^4}/\mathbb{Q}$. These are cubic twists of the original elliptic curve. I would like ...

Let $G$ be a group corresponding to a finite Coxeter system $(W,S)$. Does there exist an algorithm, which on input $(W,S)$ tells what is the minimal cardinality of a generating set for $G$? A weaker form of the question: does there exist an algorithm to check whether $G$ can be generated by 2 e...

Assume that $G$ is a (finite) abelian group and $M$ is a matroid whose ground set is $G$. Let $X$ and $Y$ be subsets of $G$, and $H$ is the stabilizer of $X+Y$. That is $X+Y+H=X+Y$. We denote the rank function of $M$ by $r$. Then can we say that $r(X+Y)\geq$ $r(X)+r(Y)-r(H)$? Or under what condit...