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Zinovy Reichstein

Publications and source records attributed to Zinovy Reichstein.

At least 19 recordsLinked to original sources
arXiv2026

Finite abelian subgroups of algebraic groups

Let $k$ be an algebraically closed field, and let $G$ be an algebraic $k$-group. We study finite abelian $k$-subgroups $A \subset G$ whose order is not divisible by the characteristic of $k$. This is a classical topic in the theory of algebraic groups going back to the work of Borel in the early 1960s. We sharpen previously known results on the structure of $A$. In particular, we show that there exists a maximal torus $T$ of $G$ such that the index $[A: (A \cap T)]$ divides the Grothendieck torsion index $t(G)$. We also show that there exists a maximal torus $T$ such that the quotient group $A/(A \cap T)$ is ``small'' in a suitable sense. As applications of these results, we (i) give a positive answer to a question of Totaro for $G$-torsors over fields $k_r = k((t_1))((t_2)) \ldots ((t_r))$ of iterated Laurent series, (ii) prove a variant of the ``hypoth\`ese optimiste'' of Tits about splitting fields of $E_8$-torsors, and (iii) show that certain torsors over $k_r$ cannot be split by the function field of a genus $1$ curve.

math.AG↗
arXiv2025

Specialization and rigidity

We describe a general method, originated by Ofer Gabber, for showing that a very general fiber in a family has certain properties. We illustrate this method with concrete examples taken from algebraic dynamics, the rationality problem for algebraic varieties, Galois theory, quadratic form theory and the theory of central simple algebras.

math.AG↗
arXiv2024

Linear system of hypersurfaces passing through a Galois orbit

Let $d$ and $n$ be positive integers, and $E/F$ be a separable field extension of degree $m=\binom{n+d}{n}$. We show that if $|F| > 2$, then there exists a point $P\in \mathbb{P}^n(E)$ which does not lie on any degree $d$ hypersurface defined over $F$. In other words, the $m$ Galois conjugates of $P$ impose independent conditions on the $m$-dimensional $F$-vector space of degree $d$ forms in $x_0, x_1, \ldots, x_n$. As an application, we determine the maximal dimensions of linear systems $\mathcal{L}_1$ and $\mathcal{L}_2$ of hypersurfaces in $\mathbb P^n$ over a finite field $F$, where every $F$-member of $\mathcal{L}_1$ is reducible and every $F$-member of $\mathcal{L}_2$ is irreducible.

math.AG↗
arXiv2024

Hilbert's 13th problem in prime characteristic

The resolvent degree $\textrm{rd}_{\mathbb{C}}(n)$ is the smallest integer $d$ such that a root of the general polynomial $$f(x) = x^n + a_1 x^{n-1} + \ldots + a_n$$ can be expressed as a composition of algebraic functions in at most $d$ variables with complex coefficients. It is known that $\textrm{rd}_{\mathbb{C}}(n) = 1$ when $n \leqslant 5$. Hilbert was particularly interested in the next three cases: he asked if $\textrm{rd}_{\mathbb{C}}(6) = 2$ (Hilbert's Sextic Conjecture), $\textrm{rd}_{\mathbb{C}}(7) = 3$ (Hilbert's 13th Problem) and $\textrm{rd}_{\mathbb{C}}(8) = 4$ (Hilbert's Octic Conjecture). These problems remain open. It is known that $\textrm{rd}_{\mathbb{C}}(6) \leqslant 2$, $\textrm{rd}_{\mathbb{C}}(7) \leqslant 3$ and $\textrm{rd}_{\mathbb{C}}(8) \leqslant 4$. It is not known whether or not $\textrm{rd}_{\mathbb{C}}(n)$ can be $> 1$ for any $n \geqslant 6$. In this paper, we show that all three of Hilbert's conjectures can fail if we replace $\mathbb C$ with a base field of positive characteristic.

math.AG↗
arXiv2024

The power operation in the Galois cohomology of a reductive group over a number field

For a connected reductive group $G$ over a local or global field $K$, we define a *diamond* (or *power*) operation $$(\xi,n)\mapsto \xi^{\Diamond n}\,\colon\, H^1(K,G)\times {\mathbb Z}\to H^1(K,G)$$ of raising to power $n$ in the Galois cohomology pointed set (this operation is new when $K$ is a number field). We show that this power operation has many good properties. When $G$ is a torus, the set $H^1(K,G)$ has a natural group structure, and $\xi^{\Diamond n}$ then coincides with the $n$-th power of $\xi$ in this group. On the other hand, we show that a power operation on $H^1(K,G)$, functorial in $G$, which we define over local and global fields, cannot be defined for an arbitrary field $K$. Our proof of this assertion relies on the results of Appendix B written by Philippe Gille. Using this power operation, for a cohomology class $\xi$ in $H^1(K,G)$ over local or global field, we define the period ${\rm per}(\xi)$ to be the least integer $n\ge 1$ such that $\xi^{\Diamond n}=1$. We define the index ${\rm ind}(\xi)$ to be the greatest common divisor of the degrees $[L:K]$ of finite extensions $L/K$ splitting $\xi$. The period and index of a cohomology class generalize the period and index a central simple algebra over $K$. For any connected reductive group $G$ defined over a local or global field $K$, we show that ${\rm per}(\xi)$ divides ${\rm ind}(\xi)$ and that ${\rm ind}(\xi)$ may be strictly greater than ${\rm per}(\xi)$, but they always have the same prime factors.

math.NT↗
arXiv2024

Essential dimension of cohomology classes via valuation theory

We give a formula for the essential dimension of a cohomology class $α$ in $H^d(K, \mathbb{Q}_p/\mathbb{Z}_p (d))$ when $K$ is a strictly Henselian field. This formula is particularly explicit in the case, where $α$ is a Brauer class (for $d = 2$). As an application of our bound with $d = 3$, we study the essential dimension of exceptional groups by examining the image of the Rost invariant.

math.GR↗
arXiv2023

Essential dimension of symmetric groups in prime characteristic

The essential dimension $\operatorname{ed}_k({\rm S}_n)$ of the symmetric group ${\rm S}_n$ is the minimal integer $d$ such that the general polynomial $x^n + a_1 x^{n-1} + \ldots + a_n$ can be reduced to a $d$-parameter form by a Tschirnhaus transformation. Finding this number is a long-standing open problem, originating in the work of Felix Klein, long before essential dimension was formally defined. We now know that $\operatorname{ed}_k({\rm S}_n)$ lies between $\lfloor n/2 \rfloor$ and $n-3$ for every $n \geqslant 5$ and every field $k$ of characteristic different from $2$. Moreover, if $\operatorname{char}(k) = 0$, then $\operatorname{ed}_k({\rm S}_n) \geqslant \lfloor (n+1)/2 \rfloor$ for any $n \geqslant 6$. The value of $\operatorname{ed}_k({\rm S}_n)$ is not known for any $n \geqslant 8$ and any field $k$, though it is widely believed that $\operatorname{ed}_k({\rm S}_n)$ should be $n-3$ for every $n \geqslant 5$, at least in characteristic $0$. In this paper we show that for every odd prime $p$ there are infinitely many positive integers $n$ such that $\operatorname{ed}_{\mathbb F_p}(\rm{S}_n) \leqslant n-4$.

math.AG↗
arXiv2022

Linear families of smooth hypersurfaces over finitely generated fields

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 the hypothesis that the characteristic $p$ does not divide $\gcd(d, n+1)$ (in particular, there is no restriction when $K$ has characteristic $0$). Moreover, we exhibit a counterexample when $p$ divides $\gcd(d, n+1)$.

math.AG↗
arXiv2022

Rational self-maps with a regular iterate on a semiabelian variety

Let $G$ be a semiabelian variety defined over an algebraically closed field $K$ of characteristic $0$. Let $Φ\colon G\dashrightarrow G$ be a dominant rational self-map. Assume that an iterate $Φ^m \colon G \to G$ is regular for some $m \geqslant 1$ and that there exists no non-constant homomorphism $τ: G\to G_0$ of semiabelian varieties such that $τ\circ Φ^{m k}=τ$ for some $k \geqslant 1$. We show that under these assumptions $Φ$ itself must be a regular. We also prove a variant of this assertion in prime characteristic and present examples showing that our results are sharp.

math.NT↗
arXiv2022

Hilbert's 13th Problem for Algebraic Groups

The algebraic form of Hilbert's 13th Problem asks for the resolvent degree $\text{rd}(n)$ of the general polynomial $f(x) = x^n + a_1 x^{n-1} + \ldots + a_n$ of degree $n$, where $a_1, \ldots, a_n$ are independent variables. The resolvent degree is the minimal integer $d$ such that every root of $f(x)$ can be obtained in a finite number of steps, starting with $\mathbb C(a_1, \ldots, a_n)$ and adjoining algebraic functions in $\leq d$ variables at each step. Recently Farb and Wolfson defined the resolvent degree $\text{rd}_k(G)$ of any finite group $G$ and any base field $k$ of characteristic $0$. In this setting $\text{rd}(n) = \text{rd}_{\mathbb C}(S_n)$, where $S_n$ denotes the symmetric group. In this paper we define $\text{rd}_k(G)$ for every algebraic group $G$ over an arbitrary field $k$, investigate the dependency of this quantity on $k$ and show that $\text{rd}_k(G) \leq 5$ for any field $k$ and any connected group $G$. The question of whether $\text{rd}_k(G)$ can be bigger than $1$ for any field $k$ and any algebraic group $G$ over $k$ (not necessarily connected) remains open.

math.GR↗
arXiv2021

The behavior of essential dimension under specialization II

Let $G$ be a linear algebraic group over a field. We show that, under mild assumptions, in a family of primitive generically free $G$-varieties over a base variety $B$ the essential dimension of the geometric fibers may drop on a countable union of Zariski closed subsets of $B$ and stays constant away from this countable union. We give several applications of this result.

math.AG↗
arXiv2021

The behavior of essential dimension under specialization

Let $A$ be a discrete valuation ring with generic point $\eta$ and closed point $s$. We show that in a family of torsors over $\operatorname{Spec}(A)$, the essential dimension of the torsor above $s$ is less than or equal to the essential dimension of the torsor above $\eta$. We give two applications of this result, one in mixed characteristic, the other in equal characteristic.

math.AG↗
arXiv2021

On the number of generators of an algebra over a commutative ring

A theorem of O. Forster says that if $R$ is a noetherian ring of Krull dimension $d$, then any projective $R$-module of rank $n$ can be generated by $d+n$ elements. S. Chase and R. Swan subsequently showed that this bound is sharp: there exist examples that cannot be generated by fewer than $d+n$ elements. We view projective $R$-modules as $R$-forms of the non-unital $R$-algebra where the product of any two elements is $0$. The first two authors generalized Forster's theorem to forms of other algebras (not necessarily commutative, associative or unital); A. Shukla and the third author then showed that this generalized Forster bound is optimal for étale algebras. In this paper, we prove new upper and lower bound on the number of generators of an $R$-form of a $k$-algebra, where $k$ is an infinite field and $R$ has finite transcendence degree $d$ over $k$. In particular, we show that, contrary to expectations, for most types of algebras, the generalized Forster bound is far from optimal. Our results are particularly detailed in the case of Azumaya algebras. Our proofs are based on reinterpreting the problem as a question about approximating the classifying stack $BG$, where $G$ is the automorphism group of the algebra in question, by algebraic spaces of a certain form.

math.RA↗
arXiv2021

A non-commutative Nullstellensatz

Let $K$ be a field and $D$ be a finite-dimensional central division algebra over $K$. We prove a variant of the Nullstellensatz for $2$-sided ideals in the ring of polynomial maps $D^n \to D$. In the case where $D = K$ is commutative, our main result reduces to the $K$-Nullstellensatz of Laksov and Adkins-Gianni-Tognoli. In the case, where $K = \mathbb R$ is the field of real numbers and $D$ is the algebra of Hamilton quaternions, it reduces to the quaternionic Nullstellensatz recently proved by Alon and Paran.

math.RA↗
arXiv2020

On the number of symmetric presentations of a determinantal hypersurface

A hypersurface $H$ in $\mathbb{P}^r$ of degree $n$ is called determinantal if it is the zero locus of a polynomial of the form $\operatorname{det}(x_0 A_0 + \ldots + x_r A_r)$ for some $(r+1)$-tuple of $n \times n$ matrices $A = (A_0, \ldots, A_r)$. We will refer to $A$ as a presentation of $H$. Another presentation $B = (B_0, B_1, \ldots, B_r)$ of $H$ can be obtained by choosing $g_1, g_2 \in \operatorname{GL}_n$ and setting $B_i = g_1 A_i g_2$ for every $i = 0, 1, \ldots, r$. In this case $A$ and $B$ are called equivalent. The second author and A. Vistoli have shown that for $r \geq 3$ a general determinantal hypersurface admits only finitely many presentations up to equivalence. In this paper we prove a similar result for symmetric presentations for every $r \geq 2$. Here the matrices $A_0, \ldots, A_r$ are required to be symmetric, and two $(r+1)$-tuples of $n \times n$ symmetric matrices $A = (A_0, A_1, \ldots, A_r)$ and $B = (B_0, B_1, \ldots, B_r)$ are considered equivalent if there exists a $g \in \operatorname{GL}_n$ such that $B_i = g^{\rm transpose} A_i g$ for every $i = 0, \ldots, r$.

math.AG↗
arXiv2020

A graph-theoretic approach to a conjecture of Dixon and Pressman

Given $n \times n$ matrices, $A_1, \dots, A_k$, consider the linear operator $L(A_1,\dots,A_k) \, \colon \; \operatorname{M}_n \to \operatorname{M}_n$ given by \[ L(A_1,\dots,A_k)(A_{k+1})= \sum_{σ\in S_{k+1}} \operatorname{sign}(σ) A_{σ(1)}A_{σ(2)} \cdots A_{σ(k+1)}. \] The Amitsur-Levitzki theorem asserts that $L(A_1, \ldots, A_k)$ is identically $0$ for every $k \geq 2n-1$. Dixon and Pressman conjectured that if $k$ is an even number between $2$ and $2n - 2$, then the kernel of $L(A_1, \ldots, A_k)$ is of dimension $k$ for $A_1, \ldots, A_k\in \operatorname{M}_n(\mathbb{R})$ in general position. We prove this conjecture using graph-theoretic techniques.

math.RA↗