Related papers: On rank in algebraic closure

Let $k$ be a perfect field of characteristic $\neq 2$. We prove that the Schmidt rank (also known as strength) of a quartic polynomial $f$ over $k$ is bounded above in terms of only the Schmidt rank of $f$ over $\overline{k}$, an algebraic…

Let $K$ be a number field and $f_1,\ldots,f_s\in K[x_1,\ldots,x_n]$ forms of odd degrees. In 1957, Birch proved that if $n$ is sufficiently large then the forms always have a nontrivial zero in $K^n$. Apart from some small degrees, the…

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…

Let $G_1, \dots, G_k$ be vector spaces over a finite field $\mathbb{F} = \mathbb{F}_q$ with a non-trivial additive character $\chi$. The analytic rank of a multilinear form $\alpha \colon G_1 \times \dots \times G_k \to \mathbb{F}$ is…

We show that if $K$ is a monogenic, primitive, totally real number field, that contains units of every signature, then there exists a lower bound for the rank of integer universal quadratic forms defined over $K$. In particular, we extend…

Let $k$ be a field, $V$ a $k$-vector space and $X$ be a subset of $V $. A function $f:X\to k$ is weakly polynomial of degree $\leq a$, if the restriction of $f$ on any affine subspace $L\subset X$ is a polynomial of degree $\leq a$. In this…

Let $D$ be an integrally closed domain with quotient field $K$. Let $A$ be a torsion-free $D$-algebra that is finitely generated as a $D$-module. For every $a$ in $A$ we consider its minimal polynomial $\mu_a(X)\in D[X]$, i.e. the monic…

Let $f$ be a homogeneous polynomial over a field. For many fields, including number fields and function fields, we prove that the strength of $f$ is bounded above by a constant multiple of the Birch rank of $f.$ The constant depends only on…

Let $K$ be a field of characteristic zero. We deal with the algebraic closure of the field of fractions of the ring of formal power series $K[[x_1,\ldots,x_r]]$, $r\geq 2$. More precisely, we view the latter as a subfield of an iterated…

Suppose that $k$ is an arbitrary field. Consider the field $k((x_1,...,x_n))$, which is the quotient field of the ring $k[[x_1,...,x_n]]$ of formal power series in the variables $x_1,...,x_n$, with coefficients in $k$. Suppose that $\sigma$…

There are many notions of rank in multilinear algebra: tensor rank, partition rank, slice rank, and strength (or Schmidt rank) are a few examples. Typically the rank $\le r$ locus is not Zariski closed, and understanding the closure (the…

Let $A$ be a residually finite dimensional algebra (not necessarily associative) over a field $k$. Suppose first that $k$ is algebraically closed. We show that if $A$ satisfies a homogeneous almost identity $Q$, then $A$ has an ideal of…

Let $R$ be a Dedekind ring, $K$ its quotient field, and $L=K(\alpha)$ a finite field extension of $K$ defined by a monic irreducible polynomial $f(x)\in R[x]$. We give an easy version of Dedekind's criterion which computationally improves…

Let $V$ be a vector space over a field $k, P:V\to k, d\geq 3$. We show the existence of a function $C(r,d)$ such that $rank (P)\leq C(r,d)$ for any field $k,char (k)>d$, a finite-dimensional $k$-vector space $V$ and a polynomial $P:V\to k$…

Let $X$ be a matrix with entries in a polynomial ring over an algebraically closed field $K$. We prove that, if the entries of $X$ outside some $(t \times t)$-submatrix are algebraically dependent over $K$, the arithmetical rank of the…

Let $\R$ be a real closed field, $\mathcal{P},\mathcal{Q} \subset \R[X_1,...,X_k]$ finite subsets of polynomials, with the degrees of the polynomials in $\mathcal{P}$ (resp. $\mathcal{Q}$) bounded by $d$ (resp. $d_0$). Let $V \subset \R^k$…

A matrix is said to have factor width at most $k$ if it can be written as a sum of positive semidefinite matrices that are non-zero only in a single $k \times k$ principal submatrix. We explore the ``factor-width-$k$ rank'' of a matrix,…

Let $k$ be a number field and $S$ a finite set of places of $k$ containing the archimedean ones. We count the number of algebraic points of bounded height whose coordinates lie in the ring of $S$-integers of $k$. Moreover, we give an…

Let $K$ be a number field, $\overline{\mathbb Q}$, or the field of rational functions on a smooth projective curve over a perfect field, and let $V$ be a subspace of $K^N$, $N \geq 2$. Let $Z_K$ be a union of varieties defined over $K$ such…

Let $K$ be a number field of degree $n$ with ring of integers $O_K$. By means of a criterion of Gilmer for polynomially dense subsets of the ring of integers of a number field, we show that, if $h\in K[X]$ maps every element of $O_K$ of…