Related papers: On a problem of A. V. Grishin

V. V. Shchigolev has proven that over any infinite field k of characteristic p>2, the T-space generated by G={x_1^p,x_1^px_2^p,...} is finitely based, which answered a question raised by A. V. Grishin. Shchigolev went on to conjecture that…

In 1988, S. V. Okhitin proved that for any field k of characteristic zero, the T-space CP(M_2(k)) is finitely based, and he raised the question as to whether CP(A) is finitely based for every (unitary) associative algebra A with nonzero…

We show that the isomorphy type of every finitely generated field $K$ with $\chr(K)\neq2$ is encoded by a \textit{\textbf{single\ha3explicit\ha3axiom}} $\istp K\!$ \textit{\textbf{in\ha3the\ha3language\ha3of\ha3fields}}, i.e., for all…

In his thesis, S. Checcoli shows that, among other results, if $K$ is a number field and if $L/K$ is an infinite Galois extension with Galois group $G$ of finite exponent, then $L$ has uniformly bounded local degrees at every prime of $K$.…

Let $ K $ be a global function field of characteristic $ 2 $. For each non-trivial place $ v $ of $ K $, let $ K_{v} $ be the completion of $ K $ at $ v $. We show that if two non-degenerate quadratic forms are similar over every $ K_{v} $,…

Asgarli, Ghioca, and Reichstein proved that if $K$ is a field with $|K|>2$, then for any positive integers $d$ and $n$, and separable field extension $L/K$ with degree $m=\binom{n+d}{d}$, there exists a point $P\in \mathbb{P}^n(L)$ which…

Let $K$ be a field of characteristic $p>0$ and let $f(t_1,...,t_d)$ be a power series in $d$ variables with coefficients in $K$ that is algebraic over the field of multivariate rational functions $K(t_1,...,t_d)$. We prove a generalization…

Let $k$ be a finitely generated field of characteristic $p>0$ and $X$ a smooth and proper scheme over $k$. Recent works of Cadoret, Hui and Tamagawa show that, if $X$ satisfies the $\ell$-adic Tate conjecture for divisors for every prime…

Fix positive integers $n,r,d$. We show that if $n,r,d$ satisfy a suitable inequality, then any smooth hypersurface $X\subset \mathbb{P}^n$ defined over a finite field of characteristic $p$ sufficiently large contains a rational $r$-plane.…

We introduce notions of nearly good relations and N-sticky modulo a relation as tools for proving that spaces are D-spaces. As a corollary to general results about such relations, we show that C_p(X) is hereditarily a D-space whenever X is…

Let $F$ be a finite field with characteristic $p > 2$ and let $G$ be the unitary Grassmann algebra generated by an infinite dimensional vector space $V$ over $F$. In this paper, we determine a basis of the $\mathbb{Z}_{2}$-graded polynomial…

We prove that Arhangelskii's problem has a consistent positive answer: if V\models CH, then for some aleph_1-complete aleph_2-c.c. forcing notion P of cardinality aleph_2 we have that P forces ``CH and there is a Lindelof regular…

Let $F$ be a finite field with the characteristic $p > 2$ and let $G$ be the unitary Grassmann algebra generated by an infinite dimensional vector space $V$ over $F$. In this paper, we determine a basis for $\mathbb{Z}_{2}$-graded…

We prove the following result. Let k be an infinite perfect field of positive characteristic and assume that strong resolution of singularities holds over k. Let R be a localization of a commutative d-dimensional k-algebra of finite type…

Let K be a valued field of characteristic p>0 with non-p-divisible value group. We show that every finite embedding problem for K whose kernel is a p-group is properly solvable.

Let $p$ be a fixed prime number, and $q$ a power of $p$. For any curve over $\mathbb{F}_q$ and any local system on it, we have a number field generated by the traces of Frobenii at closed points, known as the trace field. We show that as we…

We give a construction of a large first-order definable family of subrings of finitely generated fields $K$ of any characteristic. We deduce that for any such $K$ there exists a first-order sentence $\varphi_K$ characterising $K$ in the…

We resolve the strong Elementary Equivalence versus Isomorphism Problem for finitely generated fields. That is, we show that for every field in this class there is a first-order sentence which characterizes this field within the class up to…

Let G be an algebraic group defined over an algebraically closed field k of characteristic zero. We give a simple proof of the following result: if H^1(L, G) = {1} for some finitely generated field extension L/k of transcendence degree \ge…

Let $K$ be a finitely generated field. We construct an $n$-dimensional linear system $\mathcal{L}$ of hypersurfaces of degree $d$ in $\mathbb{P}^n$ defined over $K$ such that each member of $\mathcal{L}$ defined over $K$ is smooth, under…