Related papers: Expressing a General Form as a Sum of Determinants

We try to write (generic) elements of classical simple Lie algebras as sums of as few elements of the orbit C of long root vectors as possible. If the Lie algebra in question is sl_n or sp_n, then C consists of all rank 1 matrices in the…

For a subset $S$ of nonnegative integers and a vector $\mathbf{a}=(a_1,\dots,a_k)$ of positive integers, let $V'_S(\mathbf{a})=\{ a_1s_1+\cdots+a_ks_k : s_i\in S\} \setminus \{0\}$. For a positive integer $n$, let $\mathcal T(n)$ be the set…

We give a short and elementary proof of a theorem of Procesi, Schacher and (independently) Gondard, Ribenboim that generalizes a famous result of Artin. Let $A$ be an $n \times n$ symmetric matrix with entries in the polynomial ring…

We say that a set $S$ is $\Delta^0_{(n)}(X)$ if membership of $n$ in $S$ is a $\Delta^0_{n}(X)$ question, uniformly in $n$. A set $X$ is low for $\Delta$-Feiner if every set $S$ that is $\Delta^0_{(n)}(X)$ is also…

Let $n$ and $k$ be integers such that $1\le k\le n$ and $f(x)$ be a nonzero polynomial of integer coefficients such that $f(m)\ne 0$ for any positive integer $m$. For any $k$-tuple $\vec{s}=(s_1, ..., s_k)$ of positive integers, we define…

We generalize the $K$ matrix formulation to non-trivial non-Abelian families of 2+1D topological orders. Given a topological order $\mathcal C$, any topological order in the same non-Abelian family as $\mathcal C$ can be efficiently…

We present an elementary proof that if $A$ is a finite set of numbers, and the sumset $A+_GA$ is small, $|A+_GA|\leq c|A|$, along a dense graph $G$, then $A$ contains $k$-term arithmetic progressions.

In this paper, we investigate the question of when the equations $A^{*s}A^{s}=(A^{*}A)^{s}$ for all $s \in S$, where $S$ is a finite set of positive integers, imply the quasinormality or normality of $A$. In particular, it is proved that if…

In this paper we study some variants of Dirac-type problems in hypergraphs. First, we show that for $k\ge 3$, if $H$ is a $k$-graph on $n\in k\mathbb N$ vertices with independence number at most $n/p$ and minimum codegree at least…

We give a simple proof of a recent result of Gollin and Jo\'o: if a possibly infinite system of homogeneous linear equations $A\vec{x} = \vec{0}$, where $A = (a_{i, j})$ is an $I \times J$ matrix, has only the trivial solution, then there…

For a finite abstract simplicial complex G with n sets, define the n x n matrix K(x,y) which is the number of subsimplices in the intersection of x and y. We call it the counting matrix of G. Similarly as the connection matrix L which is…

We prove that the maximum determinant of an $n \times n $ matrix, with entries in $\{0,1\}$ and at most $n+k$ non-zero entries, is at most $2^{k/3}$, which is best possible when $k$ is a multiple of 3. This result solves a conjecture of…

For an arbitrary given $k\geq3,$ we consider a possibility of representation of a positive number $n$ by the form $x_1...x_k+x_1+...+x_k, 1\leq x_1\leq ... \leq x_k.$ We also study a question on the smallest value of $k\geq3$ in such a…

Let $S$ be a subset of a group $G$ (not necessarily abelian) such that $S\,\cap -S$ is empty or contains only elements of order $2$, and let $\mathbf{h}=(h_1,\ldots, h_m)\in \mathbb{N}^m$ and $\mathbf{k}=(k_1, \ldots, k_n)\in \mathbb{N}^n$.…

Let $\mathfrak S_{[i,j]}$ be the subgroup of the symmetric group $\mathfrak S_n$ generated by adjacent transpositions $(i,i+1), \dotsc, (j-1,j)$, assuming $1 \leq i < j \leq n$. We give a combinatorial rule for evaluating induced sign…

Let K be an algebraic number field and O_K be its ring of integers. Let S_K be the set of elements in O_K which are sums of squares in O_K and s(O_K) the minimal number of squares necessary to represent -1in O_K. Let g( S_K ) be the…

We classify Keller maps $x + H$ in dimension $n$ over fields with $\tfrac16$, for which $H$ is homogeneous, and (1) deg $H = 3$ and rk $JH \le 2$; (2) deg $H = 3$ and $n \le 4$; (3) deg $H = 4$ and $n \le 3$; (4) deg $H = 4 = n$ and $H_1,…

Given two $\left( n+1\right) \times\left( n+1\right)$-matrices $A$ and $B$ over a commutative ring, and some $k\in\left\{ 0,1,\ldots,n\right\}$, we consider the $\dbinom{n}{k}\times\dbinom{n}{k}$-matrix $W$ whose entries are $\left(…

Let $A$ and $B$ be complex numbers, and let $(w_n)_{n\ge0}$ be a sequence of complex numbers with $w_{n+1}=Aw_n-Bw_{n-1}$ for all $n=1,2,3,\ldots$. When $w_0=0$ and $w_1=1$, the sequence $(w_n)_{n\ge0}$ is just the Lucas sequence…

We consider the problem of finding the smallest rank of a complex matrix whose absolute values of the entries are given. We call this minimum the phaseless rank of the matrix of the entrywise absolute values. In this paper we study this…