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There is a famous saying in economics: When everyone pursues his or her own interests, there is an invisible hand that brings the market to equilibrium. However, this is not always the case. Here is ...

Let $Trop(Gr(m,n))$ denote the tropicalization of the grassmannian $Gr(n,m)$. Let $\phi^m : \mathbb R^{n \choose 2} \rightarrow \mathbb R^{n \choose m}$ such that $X_{i,j} \rightarrow X_{i_1,...,i_m}$...

Let $A_1 \subset A_2 \subset \cdots$ and each $A_i$ is a finite set of polynomials over variables $x_1, \ldots, x_n$. For each $i$, let $N_i$ be the Newton polytope of $A_i$. Since $A_{\infty}$ has ...

We would like to know if the following claim is true: (If you don't know the definition of a tropical hyperplane, then please consider the case when d=3) Let $P_1,\cdots,P_d$ be full dimensional (...

Let $K$ be an algebraically closed field and $K[\underline{x}]$ its ring of polynomials in $n$ variables $x_1,\cdots, x_n$. Let $J\leq K[\underline{x}]$ be an ideal such that there are no monomials in ...

Let $C$ be a convex polytope in $\mathbb{R}^n$ with $m$ extremal points. Let $p\in \{1,2\}$. Can the $\ell^p$-projection $\Pi_C:\mathbb{R}^n\to C$ $$ \Pi_C(x) \in \operatorname{argmin}_{z\in C}\, \|x-...