Related papers: Binary forms with three different relative ranks

Let $F, G \in \mathbb{Z}[X, Y]$ be binary forms of degree $\geq 3$ with automorphism groups isomorphic to the dihedral group of cardinality $6$ or $12$. We characterize exactly when $F$ and $G$ have the same value set, i.e. $F(\mathbb{Z}^2)…

Let $ {\mathbf k} $ be a field and $Q\in {\mathbf k}[x_1, \ldots, x_s]$ a form (homogeneous polynomial) of degree $d>1.$ The ${\mathbf k}$-Schmidt rank $rk_{\mathbf k}(Q)$ of $Q$ is the minimal $r$ such that $Q= \sum_{i=1}^r R_iS_i$ with…

To our knowledge at the time of writing, the maximum Waring rank for the set of all ternary forms of degree $d$ (with coefficients in an algebraically closed field of characteristic zero) is known only for $d\le 4$. The best upper bound…

In this paper, it is shown that if F(x , y) is an irreducible binary form with integral coefficients and degree $n \geq 3$, then provided that the absolute value of the discriminant of F is large enough, the equation |F(x , y)| = 1 has at…

Motivated by (and using tools from) communication complexity, we investigate the relationship between the following two ranks of a $0$-$1$ matrix: its nonnegative rank and its binary rank (the $\log$ of the latter being the unambiguous…

In this paper, we introduce the notion of K-rank, where K is an strong amalgamation Fraisse class. Roughly speaking, the K-rank of a partial type is the number of "copies" of K that can be "independently coded" inside of the type. We study…

We prove bounds for multilinear operators on $\R^d$ given by multipliers which are singular along a $k$ dimensional subspace. The new case of interest is when the rank $k/d$ is not an integer. Connections with the concept of {\em true…

Given a field $K$ and $n > 1$, we say that a polynomial $f \in K[x]$ has newly reducible $n$th iterate over $K$ if $f^{n-1}$ is irreducible over $K$, but $f^n$ is not (here $f^i$ denotes the $i$th iterate of $f$). We pose the problem of…

We present an example of a subfield $\mathcal{F}\subset\mathbb{R}$ and a matrix $A$ whose conventional and nonnegative ranks equal five, but the nonnegative rank with respect to $\mathcal{F}$ equals six. In other words, $A$ can be…

For a totally positive definite quadratic form over the ring of integers of a totally real number field $K$, we show that there are only finitely many totally real field extensions of $K$ of a fixed degree over which the form is universal…

For a finite field k and a triple of integers g \ge r \ge s \ge 0, we count the number of semilinear endomorphisms of a g-dimensional k-vector space which have rank r and stable rank s. Such endomorphisms show up naturally in the…

Let $K$ be a number field of degree $n$ over ${\mathbb Q}$. Then the 4-rank of the strict class group of $K$ is at least ${\text{rank}_2 \, } ({ E_{K}^{+} } / E_K^2) - \lfloor n /2 \rfloor$ where $E_K$ and ${ E_{K}^{+} }$ denote the units…

We introduce a new concept of rank - relative rank associated to a filtered collection of polynomials. When the filtration is trivial our relative rank coincides with Schmidt rank (also called strength). We also introduce the notion of…

Given a simple undirected graph, one can construct from it a $c$-step nilpotent Lie algebra for every $c \geq 2$ and over any field $K$, in particular also over the real and complex numbers. These Lie algebras form an important class of…

Let V be a vector space of dimension n over a field K and let Symm(V) denote the space of symmetric bilinear forms defined on V x V. Let M be a subspace of Symm(V). We investigate a variety of hypotheses concerning the rank of elements in M…

Let $F, G \in \mathbb{Z}[X, Y]$ be binary forms of degree $\geq 3$, non-zero discriminant and with automorphism group isomorphic to $D_4$. If $F(\mathbb{Z}^2) = G(\mathbb{Z}^2)$, we show that $F$ and $G$ are ${\rm GL}(2,…

Let $K$ be a field and let $V$ be a vector space of dimension $n$ over $K$. Let $M$ be a subspace of bilinear forms defined on $V\times V$. Let $r$ be the number of different non-zero ranks that occur among the elements of $M$. Our aim is…

The strength of a multivariate homogeneous polynomial is the minimal number of terms in an expression as a sum of products of lower-degree homogeneous polynomials. Partition rank is the analogue for multilinear forms. Both ranks can drop…

We study real ternary forms whose real rank equals the generic complex rank, and we characterize the semialgebraic set of sums of powers representations with that rank. Complete results are obtained for quadrics and cubics. For quintics we…

Given a relational structure M on n elements, let D(M) be the minimum quantifier rank of a first order formula identifying M up to isomorphism in the class of n-element structures. The obvious upper bound is D(M)\le n. We show that if the…