$E_f$ is an equivalence relation on $X$ and conversely for every equivalence relation $R$, you can construct the topological quotient $X/R$ and for the induced map $f:X\to X/R$, $E_f=R$. There are ...

Let $f(x) = a_nx^n+\cdots+a_0$ be a polynomial with coefficients in the ring of integers $\mathcal{O}_K$ of a number field $K$. Then for every nonzero root $\alpha$ of $f$ in $K$ one has $a_n \alpha, \...

1- Shurman - Geometry of the Quintic 2- Weeks - The Shape of Space 3- Hatcher - Topology of Numbers 4- Arnold - Real Algebraic Geometry 5- Stepanov - Arithmetic of Algebraic Curves

By the evident isomorphism of $\mathbb R^m\otimes \mathbb R^n$ with $M_{m\times n}(\mathbb R)$, the orbits of the action of $Gl(\mathbb R^m)\otimes Gl(\mathbb R^n)$ corresponds to orbits of the action ...

As noted by Todd this is a problem in projective geometry of conics. So we can use the duality principle and reformulate the "only if" part by the following statement (the other part is equivalent): ...

It suffices to prove the bound for (non-planar) polytopes $K$ with integer vertices. Let $v$ be a vertex of $K$. By triangulating faces of $K$ which does not contain $v$ and considering the ...

One can show that for every $m$ with $I_m\neq 0$, after a generic change of coordinates, $$I_m = \langle f_{m1},\dots, f_{mj_m}\rangle \oplus x_3I_{m-1}, $$ s.th. the initial terms of $f_{m1},\dots, ...

This paper is devoted to the generalization of Tannakian formalism for fiber functors over more general tensor categories: $F:\cal{C}\to \cal{D}$. One can see easily that for representing $F$ as a ...